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NPS ARCHIVE 
1997.09 
WONNACOTT, W. 






NAVAL POSTGRADUATE SCHOOL 
Monterey, California 




THESIS 



MODELING IN THE DESIGN AND ANALYSIS OF A 
HIT-TO-KILL ROCKET GUIDANCE KIT 

by 

W. Mark Wonnacott 



September, 1997 



Thesis Advisor: 
Second Reader: 



Conrad F. Newberry 
Louis V. Schmidt 



Thesis 
W757 



Approved for public release; distribution is unlimited. 



DUDLEY KNOX LIBF 

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1. AGENCY USE ONLY (Leave blank) 



2. REPORT DATE 

September 1997 



3. REPORT TYPE AND DATES COVERED 



Master's Thesis 



4. title and subtitle MODELING IN THE DESIGN AND 

ANALYSIS OF A HIT-TO-KILL ROCKET GUIDANCE KIT 



6. AUTHOR(S) 

Wonnacott, W. Mark 



5. FUNDING NUMBERS 



7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 

Naval Postgraduate School 
Monterey, CA 93943-5000 



8. PERFORMING ORGANIZATION 
REPORT NUMBER 



9. SPONSORING / MONITORING AGENCY NAME(S) AND ADDRESS(ES) 



10. SPONSORING / MONITORING 
AGENCY REPORT NUMBER 



11. SUPPLEMENTARY NOTES 

The views expressed in this thesis are those of the author and do not reflect the official policy or 
position of the Department of Defense or the U.S. Government. 



12a. DISTRIBUTION / AVAILABILITY STATEMENT 

Approved for public release; distribution unlimited. 



12b. DISTRIBUTION CODE 



13. ABSTRACT (maximum 200 words) 

This thesis presents several computer models used in the design and analysis of a Hit-to-Kill Rocket 
Guidance Kit (HRGK). The HRGK — proposed as an inexpensive add-on kit — has the potential of converting 
unguided 2.75" diameter rockets into precision weapons against non-tank targets. A Naval Postgraduate School 
design team recently participated in a nation-wide graduate student competition for the design of such a kit. The 
design and analysis process led the author to develop and use various computer models and simulations. This 
thesis documents three distinct types of computer models found useful in the design. 

The first, operational effectiveness modeling, established the cost effectiveness of the NPS HRGK. The 
second was related to the preliminary sizing of various design aspects — ensuring the proper flow-down of system 
requirements into design specifications. The third was a six-degree of freedom (6DOF) simulation, developed to 
perform detailed analyses of the HRGK's performance. 

Although the models presented in this thesis pertain to the HRGK, the basic principles apply to the design 
or evaluation of other missile systems, and this thesis provides general insights regarding the benefits and 
limitations of computer modeling in missile design. 



14. SUBJECT TERMS 

Rocket Guidance Kit, Missile Design, Missile Modeling and Simulation 



16. NUMBER OF PAGES 

106 



16. PRICE CODE 



17. SECURITY CLASSIFICATION 
OF REPORT 

Unclassified 



18. SECURITY CLASSIFICATION 
OF THIS PAGE 

Unclassified 



19. SECURITY CLASSIFICATION 
OF ABSTRACT 

Unclassified 



20. 



LIMITATION OF 
ABSTRACT 



UL 



NSN 7540-01-280-5500 



Standard Form 298 (Rev. 2-89) 



u 



Approved for public release; distribution is unlimited 



MODELING IN THE DESIGN AND ANALYSIS OF A 
HIT-TO-KILL ROCKET GUIDANCE KIT 

W. Mark Wonnacott 
B.S.M.E., Brigham Young University, 1989 

Submitted in partial fulfillment of the 
requirements for the degree of 



MASTER OF SCIENCE IN AERONAUTICAL ENGINEERING 



from the 



NAVAL POSTGRADUATE SCHOOL 
September 1997 



DUDLEY KNOX LIBRARY 

NAVAL POSTGRADUATE SCHOOL 

MONTEREY, CA 93943-5101 



DUDLEY KMOX LIBRARY 

^ADUATE SCHOOL 

,-5101 



ABSTRACT 



This thesis presents several computer models used in the design and analysis of a 
Hit-to-Kill Rocket Guidance Kit (HRGK). The HRGK — proposed as an inexpensive add- 
on kit — has the potential of converting unguided 2.75" diameter rockets into precision 
weapons against non-tank targets. A Naval Postgraduate School design team recently 
participated in a nation-wide graduate student competition for the design of such a kit. 
The design and analysis process led the author to develop and use various computer 
models and simulations. This thesis documents three distinct types of computer models 
found useful in the design. 

The first, operational effectiveness modeling, established the cost effectiveness of 
the NPS HRGK. The second was related to the preliminary sizing of various design 
aspects — ensuring the proper flow-down of system requirements into design 
specifications. The third was a six-degree of freedom (6DOF) simulation, developed to 
perform detailed analyses of the HRGK's performance. 

Although the models presented in this thesis pertain to the HRGK, the basic 
principles apply to the design or evaluation of other missile systems, and this thesis 
provides general insights regarding the benefits and limitations of computer modeling in 
missile design. 



VI 



TABLE OF CONTENTS 

I. INTRODUCTION 1 

A. BACKGROUND 1 

B. NPS HRGK DESIGN 2 

C. THESIS SCOPE AND OUTLINE 4 

H. COST OF OPERATIONAL EFFECTIVENESS ANALYSES 7 

A. SIMPLIFYING ASSUMPTIONS 7 

B. COST PER KILL 10 

C. KILLS PER SORTIE 15 

D. COST PER MISSION 15 

E. OBSERVATIONS 20 

IH. PRELIMINARY SIZING ANALYSES 23 

A. LENGTH, WEIGHT, AND CENTER OF GRAVITY 23 

B. MANEUVERABILITY 26 

C. SEEKER FIELD OF VIEW 39 

D. COMMENTS AND OBSERVATIONS 46 

IV. SIX-DEGREE OF FREEDOM SIMULATION 51 

A. SIMULATION OVERVIEW 51 

B. SIMULATION COMPONENTS 54 

C. EQUATIONS OF MOTION 55 

D. NPS HRGK RESULTS 58 

E. OBSERVATIONS 60 

V. CONCLUSIONS AND RECOMMENDATIONS 61 

APPENDIX A. MISSILE AERODYNAMICS NOMENCLATURE 63 

APPENDIX B. NOTES ON MATLAB AND SIMULINK 65 

APPENDIX C MATLAB CODE LISTINGS. 67 

LIST OF REFERENCES 89 

INITIAL DISTRIBUTION LIST 91 



vu 



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LIST OF FIGURES 

1. Sample Normal Plane View for a 12 mil, One-Dimensional Dispersion at 5 km 
Downrange 3 

2. Schematic View of the NPS HRGK Design 5 

3 . Projected Area for Light Armored Vehicles 9 

4. Single-Shot P K and Number of Rockets Required per Target as a Function of 
CEP 13 

5 . Relative Cost per Kill against Various Targets 14 

6. AH-64 Apache Attack Helicopter Sorties Required to Achieve Mission Objective. . . 19 

7 . Number of Weapons Fired to Achieve Mission Objective 19 

8 . Number of Targets Killed by Category, Weapon, and Loadout Option 20 

9 . Physical Size and Center of Gravity Design Space 25 

10. Graphical Representation of Maneuverability Model Inputs 27 

1 1 . Closed-Form Solution for Maneuverability Specifications with a Stationary Target. . 29 

12. Moving Target Scenario for Maneuverability Analysis 31 

13. Maximum Turn Rate Required to Hit Target (95% Probability) 33 

14. Minimum Turn Radius Required to Hit Target (95% Probability) 33 

1 5 . Plane- View, Free-Body Diagram for Steady-Turn Maneuver 35 

16. Required Thrust and Angle of Attack for Maximum Turn Rate or Minimum Turn 
Radius 38 

17. Required Field of View. Results for 95% Probability of Target Staying in Field of 
View with Worst Case Target Motion 40 

18. Target Distribution Model 42 

19. Illustrative Case for Probabilistic Seeker Field of View Analysis 43 

20. Possible Target Locations 44 

2 1 . Alternative Method for Computing Target Distribution Area within Seeker Field of 
View Limits 45 

22. Probability Density Functions, (a) Rocket cross- track error, (b) Target cross-track 
position, and (c) Probability of acquisition 45 

23. Probability of Target in Field of View for Maximum Launch Range (6 km) 47 

24. Sensitivity Analysis of the Steady-Turn Maneuver Model for Thruster Sizing 49 

25. HRGK 6DOF Simulation Overall Block Diagram 52 

26. Detailed View of the Rocket Dynamics Block 53 

27. Detailed View of the Parameters and Coefficients Block 53 

28. Future 6DOF Architecture 54 

29. HRGK-Equipped Rocket Ballistic Trajectories 59 

30. Sustained and Pulsed Thruster Maneuvers 59 



IX 



LIST OF TABLES 

1 . Measures of Merit and Their Associated Objectives and Requirements 3 

2. COEA Target Set with Assumed Dimensions, Desired P K , and P^ 9 

3. Single-Shot Probability of Kill Computer Model Inputs 11 

4. Cost per Kill 14 

5. Kills per AH-64 Apache Sortie against Light Armor Targets 15 

6. Weight, Length, and Center of Gravity Model Inputs 24 

7. Maneuverability Model Inputs 26 

8. Steady-Turn Maneuver Computer Model Inputs 36 

9. Seeker Field of View Analysis Inputs 41 



XI 



Xll 



ACKNOWLEDGMENT 

I am grateful for the support of my thesis advisor, Dr. Conrad Newberry, and the 
second reader, Dr. Lou Schmidt, for their help and guidance. Dr. Robert Ball, Aeronautics 
and Astronautics Department, and Major Vince Tobin and Captain Pat Mason, U.S. Army, 
were also very helpful with the cost of operational effectiveness portion of this thesis. I am 
very appreciative of the sacrifices of Major Boaz Pomerantz, Israeli Air Force; Major 
Silvino L. Silva, Brazilian Air Force; and Lieutenant Nigel A. Nurse, U.S. Navy who 
served on the NPS Hit-to-Kill Rocket Guidance Kit Design Team. 

I am indeed indebted to my management at the Naval Air Warfare Center Weapons 
Division and the NAWCWPNS Long-Term Fellowship Committee for their continued 
support of my studies. I wish to specifically acknowledge Tomma Bersie, Doug Savage, 
John Freeman, Lee Gilbert, Paul Homer, and Ron Derr for allowing me this time at the 
Naval Postgraduate School. Special thanks also go to Ted Fincher and Ken Morton for 
inspiring me to pursue a graduate education. 

But, my most tender and heart-felt gratitude is reserved for my sweet wife Diane 
and our dear children: Andrea, Jared, Nathan, Melissa, Marsha, Jennifer, and Kayla. 
Without their love and sacrifice, this thesis would not have been possible. 



xiu 



XIV 



I. INTRODUCTION 

During the first half of the 1996-1997 Naval Postgraduate School (NPS) academic 
year, a team of aeronautical engineering graduate students was assembled to perform the 
conceptual design of a hit-to-kill guided rocket kit (HRGK). (The kit attaches to existing 
unguided rockets to give them precision strike capability.) As part of the design team, the 
author developed several computer codes or models to facilitate the design and analysis of 
the HRGK. Since the completion of the design project, many of these codes have been 
expanded or refined. 

This thesis documents the theory used in the development of the improved 
computer models. It also describes the process of using the models and presents the 
modeling results for the NPS HRGK design. These descriptions provide an example of the 
type of modeling that can be used in the design and analysis of missiles. 

This introductory chapter provides background information about the HRGK and 
pertinent details of the NPS HRGK design. It also outlines the remaining chapters of the 
thesis. 

A. BACKGROUND 

In the mid 1990s, US Army Aviation identified the need for a low cost, precision 
kill weapon system for use against soft or light-armor targets (such as trucks, armored 
personnel carriers, artillery, air defense systems, command posts, amphibious landing 
vehicles, or patrol boats). The objective of such a system would be to minimize the cost per 
kill for these types of targets. The 2.75-inch rocket (Mark-66) with a unitary, high- 
explosive (HE) warhead (M151) and a point detonation fuze (M423) was subsequently 
identified as a promising candidate for this low-cost mission. 

These unguided rockets cost only about $ 1 ,000 per unit, and large inventories are 
already available to the military services. The 2.75-inch rockets are launched from 
helicopters, fixed-wing aircraft, surface combat vehicles, and naval vessels. U.S. Army 
Helicopters carry the rockets in either a 7- or 19-tube launcher pod as part of the Hydra-70 
weapon system. The Hydra-70 system includes the weapon (rocket, fuze, and warhead), 
launch pod, and an armament management system. With this system, the rockets can be 
fired in single-shot mode or in various sized ripple or salvo shots. The rockets can carry 
various warheads or payloads (including HE, kinetic energy flechettes, multi-purpose 



submunitions, illumination flare, marking smoke, and chaff) with various fuzing options 
(including remotely set timed burst and super-quick or delayed contact fuzes). 

The Mark-66 rocket with warhead and fuze is spin stabilized to provide some 
degree of accuracy. The rocket uses a fluted single throat nozzle to achieve approximately a 
10 Hz spin by the time it exits the launcher. By motor burnout, the spin rate has increased 
to approximately 35 Hz. After motor burn out, the rocket's wrap-around fins de-spin the 
projectile and cause a significant spin rate (15-20 Hz) in the reverse direction. The spinning 
characteristics of the rocket played a significant role in the NPS design as will be discussed 
in the following section. 

The baseline rocket configuration chosen for the kit (HE warhead and super-quick 
contact fuze) is an area suppression weapon. The unguided rocket has a total 12 
milliradians (mill), one-dimensional dispersion, meaning that for every thousand meters of 
down range fly out a 12 m error results (1 a or 68%). For example, a nominal 5 km range 
launch would result in a normal-plane 1 , radial miss of up to 60 m (5 x 12) — with 39% 
probability (see Figure 1). This level of accuracy is far from that required for single-shot, 
precision kills. To achieve point target accuracy, the rocket needs to be fitted with some 
type of guidance and control kit. The conceptual NPS HRGK, described in the next 
section, was designed to serve this purpose. 

B. NPS HRGK DESIGN 

The NPS HRGK was designed to meet certain requirements. The following 
subsections summarize these requirements and the pertinent aspects of the NPS HRGK 
design. 

1. HRGK Design Specifications 

The NPS HRGK design requirements were prepared by the American Institute of 
Aeronautics and Astronautics (AIAA) Missile Systems Technical Committee and were used 
for the 1996/1997 Graduate Team Missile Design Competition sponsored by the AIAA and 
Northrop Grumman Corporation [Ref. 1]. Table 1 summarizes the HRGK's measures of 
merit with their associated objectives and pertinent requirements. 



1 For the purposes of this thesis, all accuracies are in terms of the plane normal to the rocket trajectory 
rather than in the ground plane. This convention is typical for guided missiles. 



Elevation Error 
(68% of hits) 



±60 m 



Cross Track Error 
(68% of hits) 

CE: circular error 

CEP: circular error probable 



±60 m- 




90% CE 
(2.15 a) 



50% CE- 
(1.18 a) 



60mR(1a) 
(39% CE) 



-CEP 



Figure 1. Sample Normal Plane View for a 12 mil, One-Dimensional 
Dispersion at 5 km Downrange. [(12 m/ 1000 m)5000 m = 60 m (la)]. 



Table 1 . Measures of Merit and Their Associated Objectives and Requirements. 



Measure of Merit and Objective 


Specific Requirements 


Cost 

Maximize Units Bought 


• $10,000 Avg. Unit Cost (5,000 Unit LRIF) 

• Maximum $1 Billion Total 5 Year Production Buy 


Current Systems Computability 

Maximize Operability & 
Reduce Life Cycle Cost 


• Existing Rocket, Fuze, Warhead, & Launch Pod 

— 72 inch total system maximum length 

— loaded pod center of gravity between lugs 

• Laser Designator Compatibility 


Accuracy 

Provide Hit-to-Kill Accuracy 


• 0.5 m CEP 

• 60 mph target speed 


Weight 

Maximize Kits Deliverable via Airlift 


• 30 pound Total System Maximum Weight 


Maximum Range 

Maximize Platform Standoff Survivability 


• 6 km (or More) Maximum Range 
(Maneuverability at Maximum Range) 


Minimum Range 

Maximize Launch Acceptable Region 


• 1 km (or Less) Minimum Range 

(Guidance, Fuzing, and Launch Transients) 


Adverse Weather Performance 

Maximize Operability and Combat Flexibility 


• 99% Worldwide Weather Capable 


Time to Target 

Minimize Time (Improve Survivability) 


• 16 seconds or Less to 5 km Range 



' Low Rate Initial Production 



2. NPS Design Highlights 

The following paragraphs briefly describe the major features of the NPS HRGK 
design. The design team's proposal gives a more detailed description of the design [Ref. 
2]. 

The HRGK attaches over the fuze at the front end of a 2.75 inch rocket. It utilizes a 
strapped-down, four-quadrant silicon detector for semi-active laser spot homing. The 
HRGK employs three hot gas reaction jets (thrusters) to control the rocket' s attitude and 
trajectory. A solid propellant burning in an insulated pressure chamber generates the 
thruster gas. The kit uses a look-up table scheme to control the thrusters based solely on the 
seeker outputs. Fast-cycle solenoid valves open and close the "on-off" thrusters. The 
thruster nozzles are inter-spaced between three fixed, wrap-around canards. The canards 
provide increased maneuverability (by reducing the rocket's static stability) and help 
constrain any jet and air stream interaction effects. The canards also contribute significantly 
to the HRGK's roll stability. 

Thrust and radial needle rollers effectively isolate the NPS HRGK from the 
spinning motion of the rocket. Simulations show the kit's roll rate to be less than 9 degrees 
per second after the first seconds of flight. The kit's low spin rate and the simplistic pursuit 
navigation scheme eliminate the need for inertial sensors and greatly reduce the kit's signal 
processing requirements. 

As shown in Figure 2, the basic kit is 19.0 inches long. It has a diameter of 2.75 
inches (with canards wrapped in) and weighs 7.0 pounds. The NPS HRGK attaches to the 
2.75 inch rocket between the M151 high-explosive warhead and M423 point detonation 
fuze. The all-up round is 68.75 inches long and weighs 30 lb. The guided rocket is 
compatible with the Hydra-70 rocket system. 

C. THESIS SCOPE AND OUTLINE 

The NPS HRGK design was achieved and verified with the aid of several original 
computer codes and models. The codes and models developed by the author and described 
in this thesis fall into the following three major categories: 

• cost of operational effectiveness analyses, 

• preliminary sizing analyses, and 

• six-degree of freedom simulation. 



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The following three chapters provide descriptions of the modeling processes in the 
respective categories. Each of these chapters (1) includes a description of the modeling 
methodology, (2) gives example results specific to the NPS HRGK design, and (3) 
discusses observations regarding the general use of the models. The final thesis chapter 
summarizes these conclusions and provides recommendations for additional work with the 
computer models. Appendices that contain the computer code listings and additional 
background have also been included. The appendices include a brief description of missile 
aerodynamics nomenclature (Appendix A) and a discussion concerning the use of the 
computer software Matlab® and Simulink® (Appendix B). 

In general, this thesis is not intended to be read from cover to cover. Instead this 
thesis is offered as a reference to possible analysis types and modeling methods for use in 
future conceptual design of missile systems. 



® Matlab and Simulink are registered trademarks of The Math Works, Inc. 



II. COST OF OPERATIONAL EFFECTIVENESS ANALYSES 

Cost of operational effectiveness analysis (COEA) must be an important part of the 
design process. First, COEAs provide justification for the design based on operational 
effectiveness cost savings. But, just as importantly, COEA-type analyses provide the 
designer with focus and insight into the truly important aspects of the design. Basic COEA 
models with many simplifying assumptions can be used early in the process to focus 
design decisions. As the design matures, refined models can be used to evaluate major 
design iterations. 

The COEA modeling developed for the NPS HRGK provided the justification for a 
HRGK-like system. The overall objective of the HRGK was to minimize the total cost per 
kill against appropriate targets. From the analysis, guided rockets showed a decreased cost 
per target kill and increased target kills per sortie when compared to alternative weapon 
systems. Additional modeling showed that guided rockets resulted in a reduction of sorties 
and weapon costs required to achieve a specific mission objective. The final NPS HRGK 
was an attempt to provide the lowest possible cost per kill design. 

The COEA modeling discussed in this chapter provides several different levels of 
results. The low level results of the modeling include single-shot probabilities of kill and 
the number of weapons required per kill as a function of circular error probable (CEP). 
These values are then used to derive the cost per kill and the sorties per mission results. 
The number of weapons required per kill results also provide some additional insights that 
was critical to the NPS HRGK design. 

The following section describes the simplifying assumptions used in the NPS 
HRGK COEA models. Subsequent sections provide more detailed descriptions of the 
model methodologies and the results. The chapter ends with a discussion of some 
observations concerning the use of COEA modeling in design. (Listings of the computer 
codes used in the NPS HRGK COEA are included in Appendix C.) 

A. SIMPLIFYING ASSUMPTIONS 

Several simplifying assumptions were made to facilitate the early NPS HRGK 
COEA modeling. These assumptions included defining guided and unguided weapon 
trajectories and costs, establishing a target set with desired probabilities of kill (P K ) for each 



target, and estimating each target's vulnerability in terms of the conditional probability of 
kill given a hit (P^w). 

1. Weapon Accuracies 

The accuracies for the unguided rocket were taken as a function of launch range and 
were based on a 12 milliradian accuracy [Ref. 1]. As a baseline for the comparison, a mid- 
range, 3-km shot was used with a resulting 42.4 m normal-plane CEP (36 m 1- 
dimensional, standard deviation multiplied by 1.1774 — see Figure 1, for example). For the 
guided rockets, three different normal-plane CEP accuracies was assumed for comparison 
purposes — 0.5 m, 1.0 m, 1.5 m, and 3.0 m. 

For both the unguided and guided rockets, a 20° terminal dive angle (relative to the 
horizontal) was assumed as a best estimate 1 . The shallow dive caused an increase in the 
rocket's ground plane down-track error but did not affect the cross-track error. These 
effects were accounted for implicitly in determining the probability of hitting the target. 

For simple comparison purposes, the AGM-114 laser Hellfire's accuracy was 
assumed to be such that if it was shot at a target it would achieve the desired killed. 

2. Target Set 

The effectiveness analysis used four representative targets (mobile air defense unit, 
armored personnel carrier, support vehicle, and patrol boat). These targets were each 
assumed to be simple blocks of appropriate length, width, and height dimensions (see 
Figure 3). Table 2 lists the four targets with their associated dimensions [Ref. 3, 4, 5, and 
6]. The dimensions are representative for targets found in the references. Table 2 also lists 
a desired P K for each of the targets. These desired probabilities were assumed, based on the 
typical importance of the target. 

The probability of each target being hit (P H ) was computed as the joint probability 
of the rocket being within both the top-to-bottom and the side-to-side dimensions of the 
target. The projected dimensions of the targets (normal to the flight trajectory) were 
computed for both a head-on and a broad-side attack. The two attack approaches gave the 
extremes in the target's presented area as shown in Figure 3. The P w 's for the two extreme 
cases were computed and then equal-weight averaged as an estimate of the P H for uniformly 
distributed attack approaches (0°-360°). 



' Later analysis showed that unguided rockets may have even more shallow dive angles, making them 
even more inaccurate in the ground-plane. 



1 

2m 






Ss£L 7?> sn *^« 




^j\9.4m2\ 

-^ 2.5 m |__ 


'".■ 


15.0 m2 


J""^5: 








5.5 m 



Figure 3. Projected Area for Light Armored Vehicles (head-on, 
45°, and broad-side attacks with 20° dive angle). 



Table 2. COEA Target Set with Assumed Dimensions, Desired P K , and P^. 




Target Description 


Dimensions 
L x W x H 


Desired P K 


P KIH 


Mobile Air Defense Unit (Light Armor) 


5 x 2.5 x 2 m 


95% 


90% 


Armored Personnel Carrier (APC) 


5 x 2.5 x 2 m 


60% 


80% 


Support Vehicle (Large Truck) 


8 x 2.5 x 2.5 m 


50% 


100% 


Patrol Boat 


14 x 4.5 x 1.5 m 


75% 


70% 



3. Target Vulnerabilities 

For each target a "cookie-cutter" model [Ref. 7] was used in determining the single- 
shot P K (P Kss ). With the cookie cutter model, the target suffers a kill (according to an 
assumed P^) if and only if it is actually hit. "Near misses" do not provide any P K . Thus 
the value for P K was the simple product of P^ and P H . The assumed values of P^ H are 
best estimates and are shown in the last column of Table 2. These high values of P Km 
reflect the facts that (1) the targets were relatively soft, (2) the rocket is actually impacting 
the target to score a hit, and (3) the desired type of kill (firepower, mobility, catastrophic, 
etc.), which was not specified, may only require minor damage to the target. 

4. Weapon Costs and Additional P K 's 

The per unit cost of the guided rocket concepts was assumed to be $10,000 — the 
upper threshold cost from the AJAA design contest, and the unguided, unitary warhead 
round (M151) was assumed to cost $1,000 [Ref. 1]. A multipurpose submunitions 
(MPSM) round was assumed to cost $2,000 and have a 10% P K against non-tank targets. 
The Hellfire missile was assumed to have a per unit cost of $40,000 [Ref. 8] and a single- 
shot kill capability against all four representative targets (P K - 80% for tanks and P K = 
100% for all others). 



5. Additional Assumptions 

Additionally, the assumption was made that all weapons functioned during launch, 
flyout, and fuzing; with 100% reliability. The aimpoint was assumed to be the centroid of 
the target block without any bias errors in designation or flyout. It was also assumed that 
all rocket accuracy errors had a bivariate normal distribution with independent, equi- 
variance (a 2 ) distributions along any two orthogonal axes normal to the flyout. 
Furthermore, the errors were assumed independent from rocket to rocket. Thus, the CEP 
was 1.1774 a and the P K for n multiple rockets was 1 - (l-P Ksing i eshot ) D - 

The simplifying assumptions discussed in this section are reasonably valid under 
many conditions. The last section of this chapter (II.E. Observations) includes a brief 
discussion on the affect of some of the assumptions. 

B. COST PER KILL 

The overall objective of the HRGK was to rninimize the total cost per kill against 
appropriate targets. To determine the cost per kill, the probability of kill per single weapon 
had to be established using the assumptions discussed above. This single-shot probability 
of kill, P Kss , with the additional assumptions above, could then be used to determine the 
number of weapons required to achieve the desired P K . From the number of weapons 
required per kill and the assumed cost per weapon, the estimated cost per kill could then be 
established. The process for determining the cost per kill and the results for the HRGK 
analysis are discussed in the following subsections. 

1. Methodology 

The methodology for determining the cost per kill involves determining the P Kss for 
each weapon against each target and then determining the number of weapons required to 
achieve the desired P K . A computer program was written in Matlab to compute the P Kss 
and number of weapons required per kill as a function of CEP. This program, which also 
computes a relative cost per kill tradeoff plot, is described in the following paragraphs. 

a. Model Inputs 

Only a small number of inputs are required for the single-shot probability of 
kill computer model. These inputs can be divided into descriptions of the target and of the 
weapon trajectory. Table 3 lists these two types of required inputs. 



10 



Table 3. Single-Shot Probability of Kill Computer Model Inputs. 



Input Name 


Input Description 


Target 




L, H, and W 


Effective Target Length, Height, and Width (unit of length) 


Pkh 


Probability of Kill given a Hit within the Effective Target Dimensions 


PkD 


Desired Probability of Kill for the Target 


Weapon Trajectory 




dive 


Dive Angle Relative to Horizontal (degrees) — 9 


CEP min 


Minimum CEP to Be Used in Computations (unit of length) 


CEP max 


Maximum CEP to Be Used in Computations (unit of length) 



b. Single-Shot Probability of Kill 

The first step in estimating the single-shot probability of kill is to establish 
the end-on and broad-side projected edges of the target normal to the weapon trajectory. 
The side-to-side dimensions for the two cases are simply X, =W and X 2 = L, for the end- 
on and broad-side cases, respectively. The top-to-bottom dimensions are given by the 
following equations: 



K = //cos0 + Lsin0 



(end-on) 



Y 2 = tfcos0 + Wsin0 



(broad-side) 



Next, the probability of hit, P H , was determined by estimating the 

probability of the weapon striking within the projected edges of the target — the joint 

X Y 

probability of being within ±— and±— of the aimpoint, where i- 1 or 2 depending on 

the approach case. Given the assumed, independent bivariate nature of the weapon errors, 
the joint probability could be computed as the product of the two — cross-track (X- 
direction) and elevation (y-direction) — probabilities. The erf function in Matlab is 
defined as the following: 

P = -%=\e- ,2 dt, 

Vt J o 

so the joint probability described above is given by the following Matlab expression 
where the argument to erf is normalized by V2 : 



11 



X : .Y\ JX I 1 



P H \ ±-±,±-t =erf -±-f=- -erf 



K 



2 ' 2 J i, 2 V2crJ 1,2 V2<T 

and where i = 1 or 2 and cr is the CEP I \ All A. The total P w is then the equal weighted 
average of />„ and P„ . In other words, P H = (P H + P H )/2. 

With the probability of hit, P H , and the input P^ H , the single-shot 
probability of kill is estimated as the product of the two. 

P = P ■ P 

c. Number of Weapons per Kill 

With the P fc5 and the assumption that flyout errors for each weapon shot are 
unbiased and independent, the number of weapons required to achieve a desired P K can be 
calculated. 

The probability of the target surviving a single shot is given as the 
complement of P Kss , \-P Kss . The probability of surviving n, independent shots is that 
quantity raised to the nth power, (l-P Kss T- Thus, the probability of kill for n, independent 
shots is the complement to the latter probability of survival, namely: 

The number of shots, n, to achieve a desired P K can be determined by 
solving the above equation for n as is shown in the following equation. 

ln(l-P K ) 



n = 



HI-PkssY 



d. Accuracy and Relative Cost Trade-off Chart 

Knowing the number of weapons, n, required to achieve a desired 
probability of kill (P K ) against a specific target, the relative weapon costs to give equal cost 
per kill can be computed. For example, if two weapons (A) with a 0.5 m CEP or six 
weapons (B) with a 2.0 m CEP would be required to achieve a desired P K , then if the cost 
of weapon B was one-third the cost or less of weapon A , weapon B would have a lower 
cost per kill against the specified target. 

In the computer model, weapon costs are normalized relative to the cost of 
the minimum input CEP, CEP_min, weapon. Also, the number of weapons required to 
achieve a desired P K are considered only as integer values. 



12 



e. Cost per Kill Calculations 

The absolute cost per kill can easily be computed from the results of the 
computer model described above. The cost per kill for a specified target and weapon is 
simply the integer number of weapons required to achieve the desired P K against that target 
multiplied by the assumed cost of that weapon. 

2. Results 

The results of the computer model described above are shown for the targets and 
weapons described above in section II.A. Simplifying Assumptions. The results include 
both the P Kss and the cost per kill. 

a. Direct Model Outputs 

The combined single shot probability of kill and the number of weapons 
required to achieve a desired P K are shown as a function of CEP in Figure 4. Various 
assumed accuracies for the unguided rockets are also marked on the plot with their 
corresponding launch ranges. 




Figure 4. Single-Shot P K and Number of Rockets Required per Target as a Function of CEP. 
(P Kss curves slope down from upper left; number of rockets curves slope up to upper right.) 



13 



Figure 5 shows the relative weapon costs for equal cost per kill as a 
function of CEP. In this chart the weapon costs are relative to the 0.5 m CEP weapon cost. 
From the chart it appears that a weapon with a 2 m CEP (B) would have an improved cost 
per kill against all targets if the weapon cost were less than 33% of the 0.5 m CEP weapon 
(A). Weapon C, at half the cost of A, has the highest cost per kill at 4 m CEP. 




2 3 4 

Normal-Plane Circular Error Probable (m) 



Figure 5. Relative Cost per Kill against Various Targets. Lettered design points (A, B, 
and C) are shown for illustration only. 

b. Cost per Kill Results 

From the number of weapons required per target kill, the costs per kill for 
each of the weapons can be computed. The results for the HRGK analysis based on the 
single assumed guided rocket cost are shown in Table 4. 

Table 4. Cost per Kill. 





Unguided 
Rocket 




Guided 


Rocket 






Target 


0.5 m 


1.0 m 


1.5 m 


3.0 m 


Hellfire 


Mobile Air Defense Unit 


$2,219k 


$20k 


$20k 


$40k 


$120k 


$40k 


Light Armor 


$764k 


$10k 


$10k 


$20k 


$50k 


$40k 


Large Truck 


$294k 


$10k 


$10k 


$10k 


$20k 


$40k 


Patrol Boat 


$467k 


$20k 


$20k 


$20k 


$40k 


$40k 



14 



C. KILLS PER SORTIE 

A shortcoming of the cost per kill analysis previously presented is illustrated in the 
following example. A weapon with a 2.0 m CEP (such as B in Figure 5) and a quarter of 
the cost of a 0.5 m CEP weapon has a lower cost per kill than the 0.5 m CEP weapon 
(such as A in Figure 5). However, it would take three times as many of the less accurate 
(but inexpensive) weapons to achieve the same number of kills against mobile air defense 
targets. The number of weapons available for a given mission would be limited by the 
number of launch platforms sorties. The importance of kills per sortie is completely ignored 
in the cost per kill data. A kills per sortie analysis helps to illustrate the importance of 
precision in strike warfare. 

The kills per sortie analysis performed for the HRGK was based on AH-64 Apache 
attack helicopter sorties. The helicopters were assumed to have either pure or mixed 
weapon payloads or loadouts. The pure loadouts were either 16 Hellfires or 76 rockets 
(four 19-tube rocket launchers). The mixed loadouts combined eight Hellfires with a 
limited number of unitary warhead (M151) rockets (guided or unguided) and additional 
MPSM-equipped rockets. The number of kills per sortie were calculated based on the P Kss 
data in Figure 4 and the assumed P^'s from the sub-section II.A.4. Weapon Costs and 
Additional PK's. The results are tabulated in Table 5 for the light armor target case. 

Table 5. Kills per AH-64 Apache Sortie against Light Armor Targets. 



Weapon 


Maximum Load 


Mixed Load 


Hellfire 


16 Hellfires 


16 Kills 


8 Hellfires & 38 MPSM 


9.90 Kills 


HRGK (0.5m) 


76 HRGKs 


60.7 Kills 


8 Hellfires, 24 MPSM, & 14 HRGKs 


20.4 Kills 


HRGK (1.0m) 


76 HRGKs 


52.5 Kills 


8 Hellfires, 24 MPSM, & 14 HRGKs 


18.9 Kills 


HRGK (1.5m) 


76 HRGKs 


38.7 Kills 


8 Hellfires, 24 MPSM, & 14 HRGKs 


16.3 Kills 


HRGK (3.0m) 


76 HRGKs 


15.1 Kills 


8 Hellfires, 24 MPSM, & 14 HRGKs 


12.0 Kills 


Unguided 


76 Rockets 


0.09 Kills 


8 Hellfires, 24 MPSM, & 14 Rockets 


9.22 Kills 



D. 



COST PER MISSION 



A final type of modeling performed by the author for the NPS HRGK design 
combines the attributes of the cost per kill and the kills per sortie analyses. This analysis 
assesses the cost (in terms of sorties and cost of weapons fired) to achieve a mission 



15 



objective. The following subsections outline the analysis and present the results for the 
HRGK. 

1. Mission Definition 

The cost per mission analysis was based on a specific assumed mission scenario. 
This scenario defined the target matrix with a set priority on the target types as well as a 
mission objective. The scenario also defined the attacking helicopter loadouts. The assumed 
scenario in defined in the following paragraphs. 

a. Target Matrix 

The assumed target was an assembly area with the following targets types 
listed by priority and with the target quantities in parentheses: 

• Mobile Air Defense Units (8) 

• Main Battle Tanks (40) 

• Light Armored Vehicles ( 1 50) 

• Support Vehicles (250) 

The mission objective was to destroy half of the vehicles in each combatant category. 

b. Weapon Loadouts 

The assumed initial strike force was a flight of five AH-64 Apache attack 
helicopters. As additional helicopters were needed they were added to the attack force one 
at a time. Each Apache carried a nominal load of eight Hellfires and two 19-rocket launcher 
pods. Each pod (which could be divided into zones) was loaded with twelve M261 MPSM- 
equipped rockets and seven guided or unguided M151 unitary warhead equipped-rockets. 
[Ref. 9] 

The limited number of HRGK-equipped weapons (14) per helicopter was 
based on the assumption that only six to eight laser-designated flyouts could be made from 
a firing position before the helicopters would be forced to relocate. With eight Hellfires and 
14 HRGKs, three firing positions would be required for the attack. This was considered to 
be the upper limit for a deep strike mission. The MPSM rockets would not require laser 
designation, yet they would still be effective against clusters of light armor, material, or 
personnel. [Ref. 9] 

2. Example Calculation 

The following provides an example of the process used to model the mission 
engagement. The 1 .0 m CEP case is illustrated. 



16 



a. First Five Helicopters 

The weapons loadout of the first five helicopters consists of the following: 

• 40 Hellfire Missiles (each at a tank) 

• 40 x 0.8 = 32 tank kills 

• 70 HRGK Rockets (16 at air defense units— 2 per unit, 54 at light armor) 

• 1 6 — 8 air defense unit kills with P K = 95%) 

• 54 x 0.6909 = 37.31 light armor kills 

• 120 MPSM Rockets (120 x 0.1 = 12 kills divided between non-tank targets) 

• 6 light armor kills 

• 6 support vehicle kills 

In summary, after five helicopters, 8 air defense units are killed (exceeding objective), 32 
tanks are killed (exceeding objective), 43.31 light armor vehicles are killed (31.69 short of 
the objective), and 6 support vehicles are killed (exceeding objective). 

b. Sixth Helicopter 

The following delineates the use of the sixth helicopter's weapons: 

• 8 Hellfire Missiles (each at a tank) 

• 8 x 0.8 = 6.4 tank kills 

• 14 HRGK Rockets (each at light armor) 

• 14 x 0.6909 = 9.67 light armor kills 

• 24 MPSM Rockets (24 x 0.1 = 2.4 kills divided between non-tank targets) 

• 1 .2 light armor kills 

• 1 .2 support vehicle kills 

In summary, after the sixth helicopter, 8 air defense units are killed (exceeding objective), 
38.4 tanks are killed (exceeding objective), 54.18 light armor vehicles are killed (20.82 
short of the objective), and 7.2 support vehicles are killed (exceeding objective). 

c. Seventh Helicopter 

The following is for the seventh helicopter: 

• 8 Hellfire Missiles (2 — each at a tank, 6 at light armor) 

• 2x0.8 = 1.6 tank kills 

• 6 x 1 .0 = 6 light armor kills 

• 14 HRGK Rockets (each at light armor) 

• 14 x 0.6909 = 9.67 light armor kills 

• 24 MPSM Rockets (24 x 0. 1 = 2.4 kills divided between non-tank targets) 

• 1 .2 light armor kills 

• 1 .2 support vehicle kills 

After the seventh helicopter, 8 air defense units are killed (exceeding objective), 40 tanks 
are killed (exceeding objective), 71.05 light armor vehicles are killed (3.95 short of the 
objective), and 8.4 support vehicles are killed (exceeding objective). 



17 



d. Subsequent Helicopters 

For any subsequent helicopters the weapons would be used as follows: 

• 8 Hellfire Missiles (each at light armor) 

• 8 x 1 .0 = 8 light armor kills 

• 14 HRGK Rockets (each at light armor) 

• 14 x 0.6909 = 9.67 light armor kills 

• 24 MPSM Rockets (24 x 0. 1 = 2.4 kills divided between non-tank targets) 

• 1.2 light armor kills 

• 1.2 support vehicle kills 

This gives 18.87 light armor kills per sortie; therefore, the number of sorties, after the 
seventh, to reach the objective of 75 light armor kills is 3.95/18.87 = 0.21, and the total 
number of sorties required for the mission is 7.21. 

e. Summary 

After 7.21 sorties the following number of weapons have been fired with 
the indicated assumed weapon cost: 

• 57.67 Hellfires ($2.3M) 

• 100.9 HRGKs ($1.0M) 

• 173.0 MPSMs ($0.35M) 

for a total weapon cost of $3.66M. The total targets killed were: 

• 8 air defense units 

• 40 tanks 

• 75 light armor vehicles 

• 8.65 support vehicles. 

3. Results 

Figures 6, 7, and 8 illustrate the results for the HRGK mission analysis. Figure 6 
shows the number of sorties required to achieve the mission objective. With an assumed 
cost per sortie, these numbers could be directly applied to the weapon's cost per kill. The 
inclusion of a helicopter attrition rate would also make the cost per kill more realistic. Note 
that the 1 .0 m CEP rocket does not require many more sorties than does the 0.5 m CEP. 

Figure 7 shows the number of weapons fired during the mission for each of the 
different weapon cases. The total weapon cost in millions of dollars is also indicated on the 
graph. 

Figure 8 summarizes which weapons killed which targets. It is interesting to note 
that with accurate HRGKs, the more expensive Hellfire missile could be used almost 
exclusively on tanks. On the other hand, when unguided rockets were used, nearly all the 



18 



target kills were from the Hellfire, and the unguided unitary (M151) rocket only registered 
0.242 light armor and 0.474 support vehicle kills. 



Sorties Required 

Ol O Ol 






—^^m 




^ CD 


JkJkb 


■ 







0.5m 1.0m 1.5m 3.0m unguided 
Rocket Type 





Figure 6. AH-64 Apache Attack Helicopter Sorties Required to Achieve Mission Objective. 




0.5m 



1.0m 1.5m 3.0m 

Rocket Type 



unguided 



Figure 7. Number of Weapons Fired to Achieve Mission Objective. Total weapon costs 
indicated on graph in millions of dollars.. 



19 




0.5m 



1.0m 



1.5m 



3.0m 



unguided 



Rocket Type 



Figure 8. Number of Targets Killed by Category, Weapon, and Loadout Option. 



E. 



OBSERVATIONS 



Simple COEA type modeling is essential to the design process. Besides assuring 
that the design can be cost effective, the results from early analyses provide focus for the 
design team. COEA results can show general trends that might otherwise go unnoticed. 
COEA type modeling continues to be useful throughout the design process as refined- 
model analyses can help in verifying design decisions. 

An example of how unexpected information can come from a COEA study occurred 
in the NPS HRGK analysis. Figure 4 (page 13) shows a "knee in the curve" around 1 m 
CEP. That is the approximate CEP where the P^/s begin to drop off significantly and the 
number of weapons needed per kill begins a steep increase. From the figure, it appears that 
weapons achieving a CEP much smaller than 1 m are not needed against the assumed 
targets. Therefore, as a result of the COEA modeling, designs that achieve tighter than 1 m 
CEPs at significantly higher costs are eliminated from consideration. 

On the other hand, COEA results can be very sensitive to underlying assumptions. 
The possibility of invalid assumptions can make COEA modeling too unreliable for 
confident design decisions. For example, small changes in the low-level target set 
assumptions can result in important changes in P Kss values. These values, in turn, will 



20 



impact sortie rates. Likewise, high-level assumptions like the mission definition used in the 
sorties per mission analysis will also impact the final COEA results. The validity of the 
assumptions involved in both low- and high-level models must be continuously assessed. 

Specific examples of COEA sensitivities to underlying assumptions in the NPS 
HRGK models include two opposing possibilities. If the assumed target set were expanded 
to include a smaller point target, the "knee" of the P^ curve would shift to a smaller CEP. 
Then a more expensive, 0.5 m CEP weapon might become more cost effective then less 
expensive, 1 m CEP rockets. Conversely, if the centered, unbiased aimpoint assumption 
(discussed on page 10) were not valid, the general cost effectiveness advantage of the high 
precision rocket would become smaller, and guided rockets with slightly larger dispersion 
could become more cost efficient. 

Ideally, COEA studies will include sensitivity analyses to determine the range of 
results depending on changes in the assumed inputs. With these sensitivities in mind, the 
designer can use the COEA models as powerful design tools. 



21 



22 



III. PRELIMINARY SIZING ANALYSES 

Several aspects of any design require preliminary sizing. For the NPS HRGK 
design, the physical dimensions (length, weight, and center of gravity), the thruster, and 
the seeker's field of view each needed to be sized early in the design process. 

The author developed several computer codes to allow the sizing of the kit, its 
control mechanisms, and the seeker's field of view. These codes each use top-level 
requirements to generate lower-level design specifications. This flow down of 
specifications helps to assure that the final design meets the system requirements. 

The following sections describe the methodology used in developing the computer 
models and describe some results specific to the NPS HRGK design. The chapter ends 
with observations regarding the models and the flow-down of design requirements. 

A. LENGTH, WEIGHT, AND CENTER OF GRAVITY 

The HRGK design weight and center of gravity affects the overall center of gravity 
of the guided rocket. The guided rocket's center of gravity is a critical design consideration 
in that it affects the rocket's stability in flight. Additionally, the guided rocket's center of 
gravity must be such that a fully loaded launcher pod's center of gravity is kept between the 
pod's two mounting lugs. Furthermore, the length and weight of the HRGK are 
constrained by limits specified for the overall length and weight of the guided rocket. 

A Matlab code was written by the author to help visualize the relationships 
between the HRGK size and the rocket's center of gravity. The code also maps the useable 
design space bounded by the maximum length and weight as well as the launcher's center 
of gravity constraints. This code is briefly described and some results are presented in the 
following subsections. The code listing is included in Appendix C. 

1. Methodology 

The computer code used for the length, weight, and center of gravity analysis is 

based on the simple equation defining a composite center of gravity, X CG , based on the 
component weights, W t , and component centers of gravity, x CG . Namely, 



X C g ~ 






23 



a. Inputs 

The user-defined parameters in the code are either design or constraint 
parameters as listed in Table 6. Additional model parameters needed for the model include 
the weight and center of gravity locations for the live and fired rocket and for the empty 
launcher. The length of the rocket and the location of the pod's front lug are also required. 
These parameters for the Hydra-70 rocket system were available from [Ref. 10 and 11]. 

Table 6. Weight, Length, and Center of Gravity Model Inputs. 



Input Name 


Input Description 


Design Space Parameter 


Wkl 


Range of kit weights (evenly-spaced vector) (lb) 


Lkl 


Specific kit lengths to be plotted (vector) (in) 


Dkl 


Specific kit densities to be plotted (vector) (lbm/in 3 ) 


Cgkl 


Assumed kit center of gravity (fraction of kit length back from nose) 


dia 


Assumed kit diameter (in) 


Constraint Parameters 


Lmax 


Maximum total length (in) 


Wmax 


Maximum total weight (lb) 


CgL 


Pod forward Cg limit (fraction of lug spacing back from front lug) 



The kit's length is defined as the length extending forward from the tip of 
the rocket's fuze to the forward tip of the kit. The kit's center of gravity is based on that 
length; therefore, if the kit's center of gravity were behind the tip of the fuze, the value for 
Cgkl would be greater than unity. 

b. Outputs 

The computer code creates a plot of the rocket's center of gravity 1 as a 
function of guidance kit weight. The design space is marked with various curves indicating 
user specified guidance kit densities and lengths. With any two parameters defined (kit 
weight, density, length, or guided rocket center of gravity), the other two parameters are 
fixed on the plot. The usable design space is bounded by the maximum total length and 
weight as well as the loaded launcher's center of gravity constraints. The code generates the 
described plots for both the live and the fired rocket cases. 



Measured in calibers (missile diameters) from the forward tip of the guidance kit. 



24 



2. Results 

Figure 9 shows the output from the computer model for the live rocket case. The 
solid lines represent the case where the kit's center of gravity is located at the kit's defined 
mid-length. The cross-hatched markings around the lines allow for the kit's center of 
gravity to travel within the middle third of the kit's defined length. The upper edge of the 
cross-hatched regions represents the kit's center of gravity being two-thirds of the length 
back from the nose, and the lower edge is for the center of gravity at the one-third mark. 
The bold lines represent the indicated design constraints. The NPS HRGK design point (7 
pounds and 13.625 inches longer than the fuze) is marked on the plot as a reference. 



CD 




^vKwwouu^ 1 ^ .' 



0.15 lbm/in 3 



4 5 

Guidance Kit Weight (lb) 



Figure 9. Physical Size and Center of Gravity Design Space. Kit length is defined from the 
tip of the rocket fuze forward. The launcher center of gravity constraint is tighter than 
specified for illustrative purposes. 

From the plot it appears that the loaded launcher's center of gravity requirement 
(keeping the pod's center of gravity behind the launcher's front mounting lug) is not 
binding for most conditions. Indeed, for Figure 9, the constraint was tightened to keeping 
the pod's center of gravity within the middle third of the 14 inch lug spacing so that the 
constraint could even be seen in the plot. 



25 



B. 



MANEUVERABILITY 



Another computer model was developed in Matlab to defme — based on the system 
requirements — the HRGK's maximum maneuverability specifications. An additional 
computer program uses a steady maneuver to determine the angles of attack and thruster 
forces needed to achieve the specified maneuverability. This latter code sizes the HRGK's 
thrusters. The following subsections describe these computer models and their results. 

1. Turn Rate and Turn Radius Methodology 

The maneuverability analysis determines the maximum turn rate in g's or minimum 
turn radius in kilometers needed to hit a target over various launch and seeker acquisition 
ranges. The two-dimensional model used in this analysis is described in this subsection. 

a. Model Inputs 

The maneuverability model requires several user-defined inputs. These 
inputs are listed and described in Table 7 and are illustrated graphically in Figure 10. 



Table 7. Maneuverability Model Inputs. 



Input Name 


Input Description 


Launch and Seeker Acquisition Ranges 


Rmin, Rmax 


Minimum and maximum range to target at launch (km) 


n 

Iv acq 


Specific seeker's maximum target acquisition range (vector) (km) 


Unguided Errors 


Pnt 


Launcher pointing error at time of launch — la (degrees) 


B 


Ballistic unguided flyout error — la (milliradians) 


Weapon System Delays and Limitations 


G D 


Guidance delay (time after launch before guidance is possible) (sec) 


k> 


Target identification delay (from initial acquisition until start of turn) (sec) 


s F 


Straight final flight distance (terminal non-maneuvering zone) (m) 


Missile and Target Motion 


Time 


Time values for missile range and velocity time profiles (vector) (sec) 


Dist 


Missile range profile (distances corresponding to Time) (vector) (m) 


V M 


Missile velocity profile (velocities corresponding to Time) (vector) (m/s) 


V T 


Target speed (mph) 


Statistical Confidence 


sig 


Number of sigma's confidence (e.g., sig=2 gives ±2a or 95% confidence) 



26 




V T 



First Time Seeker 
Detects Target 



$ = sig ■ ^JPnt 2 + B 2 




No Guidance Possible 
(guidance time delay, Go) 

1 



Figure 10. Graphical Representation of Maneuverability Model Inputs. 



27 



The two-dimensional model assumes that the rocket flies a straight path 
toward the initial target location but with an azimuth error angle, 6. This angle is a 
statistical combination 1 of the launcher pointing error and the ballistic fly out dispersion 
error of the rocket. After both the initial guidance delay 2 , G D , and after the rocket is within 
the assumed seeker acquisition range, R , of the target; the rocket can begin a turn toward 
the target — correcting any cross-track errors due to 6 or target motion. 

A target identification delay time, t D , between the first detection of the target 
and the start of the turn can also be included in the model. The distance associated with t D , 
S D , is based on the rocket velocity at the time of first detection (or the end of guidance 
delay, whichever is later). The model also allows for a terminal non-maneuver zone — some 
radius, S F , from the target. For modeling purposes, this zone around the target forces the 
rocket to have completed its turn to the target and align itself for a final straight approach. 

b. Stationary Target — Closed-Form Solution 

With a stationary target, the inputs described above can provide a closed- 
form solution for the maneuverability specifications. This process is laid out in Figure 1 1 . 

The equations illustrated in Figure 11 provide the maneuverability 
requirements in terms of both required turn radius, r T , and turn rate, n max . (They also 
provide the solution to the minimum required seeker field of view — FOV. This part of the 
analysis will be described more fully in a later section.) 

For the ±2o unguided error case (sig = 2), the resulting maneuverability 
values (r T and n mai ) could be interpreted as the biggest turn radius or slowest turn rate that 
would still ensure a 95% probability of hitting a stationary target. In other words, any 
larger or slower turns, than those specified, would not give a 95% confidence with the 
provided inputs. On the other hand, tighter or faster turns would give a higher than 95% 
confidence. Therefore, if the system is specified to 95% confidence, the computed values 
are the maximum turn rate and minimum turn radius required. 

c. Moving Target 

The basic equations shown in Figure 1 1 can be applied to the moving target 



1 The standard deviation, a, of 6 is the square root of the sum of the variances Pnf and B 2 — 

I 2 2 

a = ~4Pnt + B — where Pnt and B are the standard deviations for the pointing error and the ballistic 
dispersion errors, respectively. The angle 6 is some user-defined multiple {sig) of a. 

2 The range associated with G D is calculated based on the user-supplied range time profile for the rocket. 



28 



n 




Field of View and Turn Rate Calculations 



x = R sin y = R cos 6 



Rd = y~ ^Racq ~ x * => V M D 



2 2 

acq ~ x ~ 

s d ^(v MD )(t d ) 




z = Jx 2 +(y-R r )' 



FOV = sin -1 - 

= --FOV 

y 2 



law of cosines: 

d 2 = (sf +r?\ = r 2 +z 2 -2r t zcos</> 



2 2 
z ~ s f 

r t = — 

2 z cos 



n 



'M 7 



max 



^00 



Figure 1 1 . Closed-Form Solution for Maneuverability Specifications with a Stationary 

Target. 



29 



scenario shown in Figure 12. The moving target scenario assumes a "worst-case" target 
motion. The target moves with a crossing velocity in the direction opposite the flyout error. 
Then, after it is detected, the target turns in at 45° making the rocket's turn tighter. 

The moving target scenario precludes a closed-form solution of the 
maneuverability specifications. However, the equations in Figure 1 1 can be used with an 
iterative step at each of the flight milestones (target detection, start of turn, and end of turn). 
The Matlab code listing in Appendix C uses this approach. The following description 
provides an example of the iterative process used in the computer code. 

After R D is calculated using the appropriate stationary target equations (see 
Figure 11), the time required to fly that distance is available from the range time profile. 
Using the time and the target speed, the new target location can be computed. The value for 
R D can then be re-calculated based on the updated target location. These iterations continue 
until the change in R D between iterations is less than some set tolerance (such as 1 m). 

2. NPS HRGK Turn Rate and Turn Radius Analysis 

This subsection discusses the maneuverability analysis performed specifically for 
the design of the NPS HRGK. The following paragraphs describe the inputs used for the 
computer model and present the analysis results. 

a. NPS HRGK Model Inputs 

The NPS HRGK analysis used the full gamut of specified launch ranges ( 1 
to 6 kilometers) and various seeker acquisition ranges (0.75, 1.0, 1.15, 2.0, 3.0, and 6.0 
km). The assumed la values for launcher pointing error and ballistic dispersion were 3? 
and 12 mils, respectively. (Using a 95% confidence level — sig = 2, the angular flyout error 
was internally computed to be 6.16°.) The target speed was 60 mph per the system 
requirements. Initial guidance was delayed as a linear function of launch range (0.5 
seconds for 1 km, up to 1.0 second for 6 km). The target identification delay was 0.625 
seconds (arbitrarily chosen to correspond to 5 laser pulses at 8 Hertz), and the terminal 
non-maneuvering distance was arbitrarily chosen to be 200 meters. 

The rocket's velocity and distance time profiles used in the analysis were 
based on an early simulation of a HRGK-equipped rocket using a ballistic trajectory. 
Specifically, the simulation used the 150°F propellant thrust profile [Ref. 1 1] and a generic, 
hemispherical-nosed guidance kit with three canards. The aerodynamic coefficients and 
derivatives used in the simulation were determined using the software code, Advanced 
Design of Aerodynamic Missiles — ADAM [Ref. 12] . 



30 



Target at Detection 
(target begins 45° turn) 



Target at 
Rocket 
Launch 



Target Position 
(when rocket 
starts turn) 



Start of 

Rocket 

Turn 




Assumptions: 

•Target has a crossing velocity 

•After target detection, target turns 45 c 



Unguided Flyout Azimuth Error 
(Initial Pointing and Dispersion) 



Launch Helicopter 



Figure 12. Moving Target Scenario for Maneuverability Analysis. 

b. NPS HRGK Analysis Results 

Based on the inputs described above, the maximum turn rate and minimum 
turn radius specifications for the NPS HRGK are shown in Figure 13 and Figure 14, 
respectively. The results are shown as a function of launch range for several seeker 
acquisition ranges. The analysis results show that as long as the seeker can initially detect 



31 



the target at 1.15 kilometers or more, the hardest g-loading maneuver for the rocket would 
be less than 15 g's and tightest turn radius would be greater than 1.83 kilometers. 

The maximum turn rate is a strong function of launch range with short range 
launches requiring higher g-loading maneuvers. This is primarily due to higher rocket 
speeds at shorter ranges. For example, after flying 0.5 km, the rocket's velocity is 1.8 
times higher than it is after coasting out to 5 km. For equal turn radii, a 1.8 factor increase 
in velocity yields a 3.2 (or 1.8 2 ) factor increase in turn rate. The guidance delay also 
increases the turn rate at shorter launch ranges. Target identification delays primarily impact 
the shorter seeker acquisition range cases — longer delays causing higher turn rates. 

Unlike the maximum required turn rate, the minimum required rum radius is 
almost independent of rocket speed 1 . Once the launch range places the target initially 
outside the seeker's acquisition range, the turn radius curves remain fairly flat with 
increasing launch range. For example, with a 1.15-km-acquisition-range seeker, the 
minimum required turn radius changes less than 59% across the spectrum of launch ranges 
(1.83 km at 1 km launch to 2.90 km at 6 km launch). Conversely, if the seeker has 
sufficient detection range to take advantage of it, the longer range launches allow 
significantly wider turns. For example, only a gentle 12.9 km turn radius is required with a 
6 km seeker acquisition range and a 6 km launch range. 

By either measure (turn rate or turn radius), the most stressing case for the 
maneuverability requirement is the short range launch, because of rocket speed and the lack 
of reaction time. The seeker's acquisition range can also significantly impact the HRGK 
maneuverability requirements — shorter range seekers forcing higher maneuverability 
requirements. Conversely, long range seekers can alleviate some maneuverability 
requirements by detecting the target earlier and allowing gentler turns. 

An additional benefit of the maneuverability analysis is that the minimum 
required seeker acquisition range is shown to be 1.15 km. With shorter acquisition ranges 
and long launch ranges, the target's cross-track position relative to the rocket (due to 6 and 
target motion) exceeds the seeker's range. For example, with a 4.1 km launch range, the 
rocket's closest unguided approach to the target is 0.75 km (2a). For longer launch ranges, 
the closest approach distance would be greater than 0.75 km; therefore, a 0.75-km- 
detection-range seeker would not have a 95% probability of detecting the target. 



1 Rocket speed indirectly affects the turn radius since it partially determines the distance the target can 
travel before it is detected or impacted. 



32 



16 



w 
3I2 

a> 

"co 
QC 

c 
§ 8 

h- 

E 
£ 
cc 4 



t ,,,,,, 




I. J 


" ♦ 
i ' ", ♦ • 

V/ *. ♦ : 

V '. ♦' 

X V *» • 

X *k • 

,..X ;...>« .'.. * ..• ; 


Seeker 
Acquisition 
Range (km) 


•""0.75 

//////// 1 _Q 

■■■■■•■■a 1.15 




Xv ! '>. '., *. 






i 1 


1 1 





3 4 

Launch Range (km) 



Figure 13. Maximum Turn Rate Required to Hit Target (95% Probability). Only with a 
seeker able to detect a target from 1.15 km or more (R acq > 1.15) can the maximum 
range requirement (6 km) be meet. 







0.75 km 



3 4 

Launch Range (km) 



Figure 14. Minimum Turn Radius Required to Hit Target (95% Probability). Seeker 
acquisition ranges are indicated with the corresponding curves. 



33 



3. Steady-Turn Maneuver Analysis 

Once maneuverability specifications are set, a steady-turn maneuver model can be 
used to size the control mechanism needed to achieve the needed turn rate or turn radius. 
The following paragraphs describe a computer code developed by the author to perform 
this task. Later a specific example for a guided rocket is presented. The Matlab code 
listing with the NPS HRGK parameters is included in Appendix C. 

a. Mathematical Steady-Turn Model 

The purpose of this mathematical steady-turn model is to determine the 
thrust force and angle of attack (or sideslip angle) necessary to generate a constant 
acceleration turn in the horizontal. Throughout the description, the more familiar 
longitudinal nomenclature (loosely associated with the vertical plane) is used even though 
the turn is described in the directional case (loosely associated with the horizontal plane). 
The interchange of nomenclature is justified since, for the guided rocket or missile case, the 
longitudinal and directional dynamics are equivalent. A useful listing of missile 
aerodynamic nomenclature is included in Appendix A. 

Figure 15 shows a free-body diagram for a horizontal maneuver. For a 
constant acceleration turn, the sum of the forces and moments can be expressed by the 
following equations: 

£ Forces = « W = N + T = qAC Na a + kT 



( 



1 



A 



2 & 



pV 2 AC„ 



a + kT 



£ Moments = -NX CP + TX T - M q = 

= -{qAC N a)X cp + kTX T - qAd{c Mq ^j 

= -{qAC Na a)x cp + kTX T - qAd 

= -(qAC Na a)X cp + kTX T - qAd[c Mq ^j 



, Mq 2V , 



- a 



1 ^ 1 

—pV 2 AC Na X CP + TkX T -—pAC M d 2 ng 



34 



where k is a thrust multiplication factor (accounting for misalignment of the thruster and the 
effect of jet and free air steam interaction) and the other terms are either defined by Figure 
15 or are standard terms defined in Appendix A. The moments equation uses the fact that 



r T 



V 



r V 2\ 



\ a zJ 



V " V 




Figure 15. Plane- View, Free-Body Diagram for a Steady-Turn Maneuver (horizontal, left 
turn — gravity force vector pointed into the page). 

The sum of forces and sum of moments equations can be rewritten as the 
following matrix equation: 



1 



2£ 



pV 2 AC„ 



KA-t 



\ 



2*o 



pV 2 AC„ 



L C/> 



a 



4*o 



nW 



pAC M d 2 ng 



which gives the thrust magnitude and angle of attack in terms of guided rocket parameters 
and the turn rate n. Using the relationship r T = V 2 /ng, the matrix equation can be expressed 
in terms of the turn radius as the following: 

V 2 W 



1 
2g 



pV'AC„ 



kX 7 



2£ 



pV 2 AC. 



L c/> 



a 



4r T g 



gr T 



pV 2 AC M d 2 



35 



b. Steady-Turn Computer Code 

The author has developed a computer code to implement the mathematical 
model described above. This code solves for the thrust and angle of attack required to 
achieve the required turn rate or turn radius over a range of Mach numbers. The code 
requires the user-supplied inputs listed and described in Table 8. 

Table 8. Steady-Turn Maneuver Computer Model Inputs. 



Input Name 


Input Description 


Scenario Definition 


n or r T 


Desired turn rate (g's) or turn radius (ft) 


type 


Indicator for computations based either on turn rate (1) or turn radius (2) 


alt 


Pressure altitude (ft) 


Mmin, Mmax 


Minimum and maximum Mach number for the calculations 


Rocket Configuration 


Wght 


Rocket total weight at maneuvering time (lb) 


Dia 


Rocket diameter (for reference area and length) (in) 


Xcg 


Rocket center of gravity (calibers from nose) 


Xt 


Thruster location (calibers from nose) 


Aerodynamic Parameters (functions of Mach) 


Mach_dat 


Mach values for corresponding aerodynamic parameters (vector) 


CNa_dat 


Rocket normal force curve slope (corresponding to Mach) (vector) (rad 1 ) 


Xcp_dat 


Center of pressure (corresponding to Mach) (vector) (calibers from nose) 


CMq_dat 


"Pitch" damping (corresponding to Mach) (vector) (rad 1 ) 



c. NPS HRGK Example 

The following provides example solutions to the equations previously 
developed for the steady-turn model. The examples are specific to the NPS HRGK with the 
guided rocket's maneuverability specifications set at 15 g's turn rate or 1.83 km (6,000 ft) 
turn radius. The manual calculations are for the NPS HRGK design at Mach 1.0. These 
calculations are done first for the turn rate specification and then for the turn radius 
specification. The aerodynamic coefficients are from ADAM [Ref. 12]. All computations 
are for 500 feet pressure altitude and use the fired rocket center of gravity (most 
maneuvering takes place after the 1.0 sec rocket motor burn). Because no additional data is 
available, the value of k (an adjustment factor for the thruster force) is assumed to be unity. 



36 



The manual calculations are followed by sample output from the steady-turn computer 
code. 

( 1 ) Sample Manual Calculation Based on Maximum Turn Rate. 



(0.0754^)(lll4f) 2 (0.0412 ft 2 )(20.05 rad 1 ) 

z r L - L ibf s 2 / 

(0.0754 ^)(ll 14 f) 2 (0.0412 ft 2 )(20.05 rad 1 ) 



1.79ft 



Z r Z,Z Ibf s 2 / 



15 (22.8 Ibf) 



0.0824 ft 



a 



(0.0754 ^)(0.04 12 ft 2 )(-5520 rad _1 )(0.229 ft) 2 15 (32.2 f) 



4(32.2^) 



The solution to the set of equations is: 



1 


1200^" 


~T 




' 342 Ibf 




~T~ 




" 13.2 Ibf " 




"13.2 Ibf 


1.79 ft 


_QQ 1 'W ft 
771 rad _ 


a 




-3.38 Ibf ft 


— > 


a 




0.273 rad 




15.6° 



1 1200^ 



1.79 ft -99.1^ 



(2) Sample Manual Calculation Based on Minimum Turn Radius. 

(1114^) 2 (22.8 Ibf) 
(32.2^-)(6000 ft) 

(0.0754 ^)(H14f) 2 (0.0412 ft 2 )(-5520 rad _1 )(0.229 ft) 2 



4(32.2 «-)(6000 ft) 



The solution to the set of equations is: 



1 


1200^ " 


~T~ 




" 147 Ibf 




~T 




" 5.68 Ibf " 




"5.68 Ibf 


1.79 ft 


QQ 1 Ibf ft 

77,1 rad . 


a 




-1.45 Ibf ft 


— > 


a 




0.117 rad 




6.71° 



37 



(3) Sample Output from Steady-Turn Computer Code. Figure 
16 shows the output from a computer code developed by the author. The code computes 
the solution to the steady-turn equations that were manually solved in the previous 
paragraphs. 




CD 

cc 

Q. 
CO 

-a 
o 

O" 

CD 
DC 



1 


3-4 0.6 0.8 1.0 


1.2 


1.4 


1.6 


Af\ 










4U 
20 


3^ J 














0-4 0.6 0.8 .1.0 1.2 

Mach Number 



1.4 



1.6 



Figure 16. Required Thrust and Angle of Attack for Maximum Turn Rate (15 g's — 
dashed line) or Minimum Turn Radius (1.83 km — solid line) (NPS HRGK design). 

From the data plotted in Figure 16, the difference in the two 
methods of specifying maneuverability requirements is apparent. For example, at subsonic 
speeds, the model predicts that the rocket could only achieve a 15 g maneuver with very 
high thrust and extremely high angle of attack. On the other hand, the requisite 1.83 km 
turn radius can be achieved at subsonic speeds with a moderate angle of attack and low 
thruster force. 

Fortunately, 15 g maneuvers are not required when the missile has 
coasted down to subsonic speeds (see long range side of Figure 13, page 33). Clearly, the 
reasonable curve for sizing the thruster is the rninimum turn radius curve (solid line). This 
seems reasonable since the turn radius requirements remained fairly flat over the entire 
gamut of launch ranges (see Figure 14). 



38 



Designing to the minimum turn radius and given the fact that the 
rocket has slowed down to nearly Mach 1.3 by the first kilometer of flight, the thruster can 
be sized to provide 10 to 15 pounds of thrust. A design with higher thrust early in the flight 
would be ideal, since the short-range, high-speed scenario is the most demanding. 

4. Probabilistic Approach 

The maneuverability analysis performed in the previous subsections was based on 
the statistical (probabilistic) fly out of the rocket but a deterministic, worst-case target 
motion. Another approach to determining the needed level of maneuverability would be to 
apply a statistical model to the target motion as well as the fly out. This latter approach 
would give valuable insight into the reasonableness of the worst-case approach taken 
above. This type of purely probabilistic approach was needed for the seeker field of view 
analysis, which is described next. 

C. SEEKER FIELD OF VIEW 

The seeker's field of view 1 was another early consideration for the NPS HRGK 
design. Again, this design specification could be derived from top-level system 
requirements and simple mathematical models. A possible approach to determining the 
needed seeker field of view was alluded to in the maneuverability analysis section. This 
section will briefly discuss that approach and its results. Then a probabilistic approach with 
its results will be discussed in greater detail. 

1. Analysis with "Worst-Case" Target Motion Model 

The two-dimensional scenario described in subsection EQ.B.l. Turn Rate and Turn 
Radius Methodology provides a means for calculating the needed HRGK seeker field of 
view. The equations shown in Figure 1 1 (page 29), can be used to determine the field of 
view that keeps a stationary target within the seeker's field of view with some specified 
level of confidence (for example, 95%) up until the target can be detected and a turn toward 
the target can be initiated. As discussed in the earlier subsection, the equations shown in 
Figure 1 1 can be solved iteratively to determine the required field of view for a moving 
target (see Figure 12, page 31). 



1 The seeker's field of regard — meaning the total area covered by a fixed or scanning seeker — is the actual 
consideration, but for a strapped-down seeker, the field of view and field of regard can be used 
interchangeably. This thesis uses the term field of view. 



39 



Unfortunately, the target motion model used for the maneuverability analysis is a 
worst-case example. (The target crosses at maximum velocity in the opposite direction of 
the statistical flyout and pointing errors. Furthermore, after detection, the target turns in at 
45° which creates the need for a still larger field of view. See Figure 12.) 

The required field of view results shown in Figure 17 for the NPS HRGK were 
obtained using the worst-case target motion model and the computer code inputs discussed 
in subsection III.B.2. NPS HRGK Turn Rate and Turn Radius Analysis (page 30). The 
values are for half field of view and are shown as a function of launch range and for 
various seeker acquisition ranges. For example, with a maximum launch range and a seeker 
acquisition range of 1.5 km, a ±48° seeker field of view would be needed to assure (95%) 
that the target stays in the field of view up until a turn can be initiated. (With a stationary 
target, only a ±29° seeker field of view would be needed.) 




Racq = 6 km 



3 4 

Launch Range (km) 



Figure 17. Required Field of View Results for 95% Probability of Target Staying in Field 
of View with Worst Case Target Motion. (Corresponding field of view requirements for a 
stationary target are indicated with dashed lines.) 

From the curves in Figure 17, it is clear that longer launch ranges require larger 
fields of view (after longer fly outs, the rocket's cross-track error and the target's motion 



40 



away from the rocket will be larger). Shorter acquisition range seekers also require larger 
fields of view. 

Unfortunately, the indicated field of view sizes for long range launches and short 
range seekers are unattainably large, especially for a strapped-down, semi-active laser spot 
homing seeker like that on the NPS HRGK. The worst-case target motion model is 
unacceptable, and so, a probabilistic approach must be taken. 

2. Analysis with Probabilistic Target Motion Model 

The probabilistic target motion model used for the seeker field of view analysis is 
based on a distribution of target positions. In this analysis, the probability of the seeker 
acquiring a target is derived from an assumed seeker field of view and the probabilistic 
distributions for the rocket and the target. The methodology for defining a seeker field of 
view specification with this approach is developed in this subsection. The resulting 
specifications for the NPS HRGK are also presented. The Matlab code developed by the 
author to perform this analysis is included in Appendix C. 

a. Computer Model Inputs 

The inputs required for the probabilistic seeker field of view analysis are 
listed and described in Table 9. 



Table 9. Seeker Field of View Analysis Inputs. 



Input Name 


Input Description 


Seeker Descriptions 


FOVmin, max 


Minimum and maximum seeker half field of views (deg) 




Specific seeker's maximum target acquisition range (vector) (km) 


Unguided Errors and Weapon System Delays 


Pnt 


Launcher pointing error at time of launch — la (degrees) 


B 


Ballistic unguided flyout error — lo (milliradians) 


G D 


Guidance delay (time after launch before guidance is possible) (sec) 


t D 


Target identification delay (from initial acquisition until start of turn) (sec) 


Missile and Target Motion 


RL 


Range to target at launch (single value) (km) 


Time 


Time values for missile range and velocity time profiles (vector) (sec) 




Missile range profile (distances corresponding to Time) (vector) (m) 


Vx V-r 
" Tmin' * Tmax 


Minimum and maximum target speed (mph) 



41 



b. Target Motion Model 

The probabilistic seeker field of view analysis uses a simple distribution 
model for target motion. The target's speed is assumed to be constant and uniformly 
distributed between the user-supplied inputs, V Tmin and V Tmax . Its bearing, <f), is also 
assumed to be constant and to be uniformly distributed between 0° and 360°. Figure 18 
shows a plane view of possible target locations after time, t T (the flyout time before the 
rocket starts its turn to the target). The figure also includes the corresponding probability 
density function for the target's cross-track position, x. This function provides the 
probability of the target being at some specific cross-track position, x T , or between two 
cross-track limits, x, and x 2 . (The v-axis indicates the direction of the rocket flyout.) 



original 

target 

location. 



trVm 



illustrative target location 




Pt(xt) 
(probability: target 
at cross-track Xi) 



iPr{x) 



tr Vjmax 




area of possible 
target locations 



(b) 



Pt (x u xs) 
(probability: target 
between X2 and X2) 



Figure 18. Target Distribution Model, (a) Plane view of possible target positions and (b) 
Cross-track probability density function, Pj(x) 

c. Analysis Methodology 

The methodology used to compute the probability of the target being within 
the HRGK field of view, has previously been documented by the author. The following 
paragraphs briefly summarize the methodology. In these paragraphs, the assumption is 
made that if the target is within the HRGK's field of view, it will be acquired; therefore, the 
desired quantity is the probability of acquisition, P acq . The referenced technical 
memorandum should be consulted for a more complete description of the theory and the 
limitations of the simplifying assumptions. [Ref. 13] 

Figure 19 shows a sample case for the probabilistic model and shows 
several of the variables used in the model computations. To obtain the probability of the 
target being within the seeker field of view (x, < x T < x 2 ), the independent probabilities of 



42 



the rocket and target cross-tracks must be computed. The rocket cross-track computation 
assumes a normal distribution of 9 — which depends only on the independent, normally 
distributed pointing and ballistic flyout errors, Put and B, and the target's position depends 
on the target motion model described above. The cross-track distribution for the rocket is 
evaluated for the assumed range to detection (R D ), and the target distribution is evaluated 
for the assumed time at start of turn (f^ time required for the rocket to fly to R T ). 




Figure 19. Illustrative Case for Probabilistic Seeker Field of View Analysis. The rocket is 
shown at the assumed position R D . 



The probability of the rocket being at cross-track position x R is 

1 



p*w = 



-x 2 
2a 2 



<7V2tt 



where a = R D tan0 = R D ^]tan 2 (Pnt) + B 2 . Because of the assumed target distribution, the 
target has equal probability of being anywhere in the possible region (A TOT ) shown in 
Figure 20. Therefore, the probability of the target being within the field of view limits (jc ; 
and x 2 ) ] — the shaded area, A, in the figure — is the ratio of the seeker acquired area, A, to 
the total possible area, A T0T . The seeker acquired area, A, can be expressed as the area 
between the limits (* ; and x 2 ) and inside the outer circle (radius = r 2 ) minus the area 
between the limits and inside the inner circle (radius = r,). The total possible area, A T0T , is 
K { r 2 ~ r \\> tnus ' tne desired probability is given as: 



The field of view limits are x, = x R - F and x 2 = x R + F where the offset, F = R acq sin(FOV). 



43 



r T yX l ,X 2 ) - 



"n 



OT 



sign(x 2 )min( r 2 \x 2 \) sign(x 2 )min( r l ,\x 2 \) 

2 j- s jr 2 -x 2 dx - 2 j^r 2 -x 2 dx 

sign(x, ) min(r 2 ,| x x \ ) sign(x^ ) min( r, ,| x, | ) 



where the limits of the two integrations are set to never exceed the radius of the outer or 
inner circle, respectively. 



possible target locations 
within field of view 



n = tr V T > 



min 




Atot 



r2= trVjmax 
all possible target locations 



Figure 20. Possible Target Locations. The ratio of the area (A) bounded by 
the field of view limits — x, and x 2 — and the total possible area (A TOT ) is the 
probability of the target being within the two limits. 

An alternative calculation of the seeker acquired area, A, is illustrated in 
Figure 21. The calculation method relies on the fact that since the probabilities are 
sy metrical about the y-axis, the calculations only need to be made for positive values of x R ; 
thus x 2 will always be positive. The method involves calculating the area, A , , bounded by 
the circles, the lower field of view limit (x 2 ), and the y-axis as given by the equation: 

/] =min(r 2 ,£2) L 2 =nuB(,r 1 ,x 2 ) 

A, =2 j^jr 2 2 -x 2 dx-2 j^jr 2 -x 2 dx 



( 



= L,Vr 2 2 -Z 1 2 +r 2 2 sin- i p -Z^^-^+l 



(i \ 



2 sin- ! 



V r 2 



Li 



K r \J 



Next, the area A 2 is calculated. This area is bounded by the circles, the lower field of view 
limit (jc,), and the y-axis, and is given by the equation 



44 



l } =min(r 2 ,| j||) L t =min(r, ,| X| | ) 

A 2 = 2 \^r 2 -x 2 dx- 2 \ -^r 2 - x 2 dx 



= L^^jr 2 -L] + r 2 sin ' — - L 4 s\r 2 - L 2 4 + r 2 sin ' 



' L ^ 



Finally, the area A 2 is subtracted from A, if jc, is greater than zero (Figure 21 (a)) or added 
to A, if*, is less than zero (Figure 21 (b)) — A = A, - sign(x,) A 2 . 









£z 




*<&& 






^/r 


® 


;/ * 


(b)^ 


1 


r / / 


r"Xi 



Figure 21. Alternative Method for Computing Target Distribution Area within Seeker 
Field of View Limits. (A = A, - sign(x,)A 2 .) 

Now, the probability of the rocket being at a specific cross-track and the 
target being within the HRGK's field of view can be expressed as the joint probability 
P acq (x R ) = P(x R , x,<x T < x 2 ) = P R {x R )P 1 {x 1 ,x 2 ). The total probability of the rocket having 
the target in its field of view for any flyout is the integral of this joint probability. These 
probabilities are illustrated in Figure 22. The actual method for computing the probability of 
acquisition is described in the following paragraphs. 




min. target 
position, -rz 



Extreme values 
at x = ±(i2+F) 




Pacq(Xfi) = 
Pfl(Xfl)-Pr(Xi,X 2 ) 



• acq\X) 

•acq = J 'acq \X) 



(c) 



Figure 22. Probability Density Functions, (a) Rocket cross-track error, (b) Target cross- 
track position, and (c) Probability of acquisition. 



45 



d. Computer Code Description 

The Matlab computer code developed to determine the probability of target 
acquisition 1 includes a Matlab script file and a Matlab function. (Appendix B briefly 
describes the difference between script files and functions.) The script file sets the user 
inputs, determines the rocket's cross-track error standard deviation, sets the seeker offset 
(F), calculates the integral of the joint probability \P R {x) P j{x±F)dx from zero to an upper 
bound, and plots the results. The Matlab function is called by the integration scheme in the 
script file and returns values for the joint probability (P acg (x) = P /? (x)P 7 (;t±F)). These 
functions are all performed over a range of fields of view and for several seeker acquisition 
ranges. Both the script file and the function are included in Appendix C. 

e. NPS HRGK Results 

The computer code results for the NPS HRGK design case are shown in 
Figure 23. The results are for the maximum launch range case (6 km) as this is the most 
stressing case. All other input parameters are set as in previous analyses. The target speed 
is uniformly distributed between 20 to 60 miles per hour. The results show that a 95% 
probability of having the target in the field of view at time, ? p can be obtained with fields of 
view slightly smaller than those that would be required for the same results against 
stationary targets. For example, with a seeker acquisition range of 1.5 km, a ±25° seeker 
would provide a 95% probability of acquisition against the probabilistic target model. But, 
against a stationary target, a ±29° seeker would be required for the same confidence (Figure 
17, page 40). The probabilistic moving target model results in a smaller field of view 
because of the possibility of the target moving in the same direction as the rocket cross- 
track error is allowed. This possibility increases the probability of having the target within 
the field of view, even with a slightly smaller field view. 

D. COMMENTS AND OBSERVATIONS 

In concluding this chapter on preliminary sizing analyses, several observations are 
appropriate. These include (1) a discussion of the benefits and data that came from the 
computer models, (2) a word of caution about the limitations of the models, and (3) a 
listing of additional work that could be done to expand the usefulness of the preliminary 
sizing models. 



Actually the probability of the target being within the cross-track limits of the seeker's field of view. 



46 




Figure 23. Probability of Target in Field of View for Maximum Launch Range (6 km). 

1. Additional Data from Modeling 

The primary purpose of the computer models discussed in this chapter was to give 
the designer a better understanding of how the system requirements affect different design 
specifications. However, the codes not only facilitate the "flow-down" of requirements, but 
they also provide additional insights into the design of the system. In the case of the NPS 
HRGK design, several observations were made from the use of the preliminary sizing 
codes. The following paragraphs site some examples. 

From the length, weight, and center of gravity analysis it was determined early in 
the design that the launcher center of gravity constraint is not binding for any design within 
the maximum weight and length constraints. Thus, the kit's center of gravity can be placed 
to provide the best combination of performance and packaging without concern for the 
launcher's center of gravity. In a general case, a designer can use this type of knowledge to 
move through design iterations without making calculations regarding the non-binding 
constraints. Then, as the final design is approached, calculations can be made to verify the 
preliminary assessments. 

Another good example of additional information gained from the models is the 
minimum required seeker acquisition range. The required seeker range was inadvertently 



47 



discovered by trying several seeker acquisition ranges in the maneuverability analyses. The 
final seeker range specification could be derived based on any of the three target motion 
types (stationary, worst-case, or uniformly distributed). 

The maneuverability and field of view analyses provided other general insights into 
the effect of the system requirements. For example, from the analyses it became apparent 
that the minimum launch range requirement defines the maneuverability specification and 
that the maximum launch range requirement drives the seeker field of view specification. 
This insight is useful in determining design tradeoffs with relaxed system requirements. 
For example, if the minimum range requirement for the NPS HRGK were increased to 2 
km, the effect on the seeker would be minimal, but the affect on the thruster design would 
be significant. Knowing this the designer can determine the value of relaxing the system 
requirement. 

2. Limitations of Models 

Despite the usefulness of the codes discussed in this chapter, some caution must be 
applied when using these and other mathematical or computer models. Model outputs are 
sometimes skewed or erroneous because of (1) the nature of the model with its simplifying 
assumptions or (2) because of the model's sensitivity to incorrect inputs. Examples of both 
these error sources are abundant in the NPS HRGK analyses. 

For example, both the maneuverability and the field of view analyses used a two- 
dimensional model for a three dimensional problem. Implied in these models is the 
assumption that the azimuth-related maneuverability and field of view requirements 
outweigh the elevation-related requirements. In reality, the HRGK must execute significant 
maneuvers in both yaw and pitch (at the same time) to prosecute a target as it appears in the 
seeker's field of view. Furthermore, the field of view sizing may well be defined by the 
requirement to look down at a target rather than the requirement to look to the left or right 
for a target. These possibilities are not accounted for in the two-dimensional model. 

Another example of potentially incorrect modeling concerns the appropriateness of 
the target motion model. The constant heading and speed with uniform distributions may be 
an appropriate target motion model for short time of flight attacks against sluggish ground 
targets. However, this type of target motion model would not be appropriate in a surface- 
to-air scenario where the target may have evasive maneuver capability. In general, any 
target motion model must allow for a reasonable, worst-case evasive maneuver. 



48 




I I I 




' ''-,!.. ' 



0.4 0.6 0.8 1.0 1.2 1.4 1.6 

Mach Number 



0.4 0.6 0.8 1.0 1.2 1.4 1.6 

Mach Number 



Figure 24. Sensitivity Analysis of the Steady-Turn Maneuver Model for Thruster Sizing, 
(a) C M ± 100% and (b) X CP ± 3%. 

Finally, Figure 24 illustrates the importance of sensitivity analysis and the potential 

for erroneous results due to incorrect inputs. The figure shows data from the control 

thruster sizing analysis with variations in the steady-turn model parameters. The plot on the 
left (a) shows the effect of ±100% changes in C M . Doubling or zeroing the parameter 

produces only minor changes in required control thrust. Clearly, the thrust required is very 
insensitive to changes in C M . On the other hand, the plot on the right shows the impact of 

±3% changes in x CP . The small changes in this input parameter affect the required control 
thrust significantly. 

Another point of interest illustrated in the above example is the importance of 
parameter selection. In the NPS HRGK case, the parameter x CP is measured from the nose 
of the guided rocket and varies between 28.5 and 31.8 inches as a function of Mach 
number. Small percentage changes in x cp result in large percentage changes in the rocket's 
static margin (the small distance between the x cp and the x CG ). In fact, the ±3% change in 
x cp produces between a ±22.7% and ±86.5% change in static margin. Although either x CP 
or static margin could be used as model inputs, the model sensitivity to percent changes in 
the two parameters would vary significantly. 

3. Additional Work 

The computer models developed by the author could be modified to produce new or 
refined models. The addition of probabilistic target motion to the turn rate and turn radius 
code would provide useful data. The analysis of elevation plane considerations would also 
be helpful in the guided rocket design. A model that could estimate both the required 
vertical field of view and pitch axis maneuverability would help in verifying the thruster 



49 



50 



IV. SIX-DEGREE OF FREEDOM SIMULATION 

This chapter addresses the use of detailed computer modeling in the missile design 
process. The chapter describes a six-degree of freedom (6DOF) simulation developed by 
the author for the analysis of the NPS HRGK. The HRGK 6DOF is a major modification 
to a simulation developed by Professor Robert (Gary) Hutchins, Electrical and Computer 
Engineering Department, Naval Postgraduate School, and his 1996 "Navigation, Missile, 
and Avionics Systems" class. The simulation was developed in Simulink using Matlab 
functions to perform the simulation's computations. The listings of the Matlab functions 
are included in Appendix C. 

Currently, the 6DOF code requires more than 20 minutes to simulate a 15 second 
rocket flight 1 . To speed up the simulation, some of the simulation features are not yet fully 
implemented, but the author is in the process of converting the simulation to a more 
efficient computation structure. This structure uses the native Simulink "S-function" and 
was introduced to the author by Mr. Allen Robins; Dynamics and Controls Section, Naval 
Air Warfare Center Weapons Division, China Lake, California. (Appendix B briefly 
describes the different computation structures the can be used in Simulink.) Preliminary 
tests by the author have shown that the use of the S-function structure can cut simulation 
run times by a factor of nearly 30. 

This chapter begins with an overview of the simulation architecture and then 
addresses the functions of the major simulation components. A section of the chapter is 
devoted to the development of the guided rocket equations of motion. And another, 
presents simulation results used in the design of the NPS HRGK. The chapter ends with a 
few observations concerning the use of detailed simulations in the missile design process. 

A. SIMULATION OVERVIEW 

The HRGK 6DOF simulation is designed to be used with an initialization script file. 
This Matlab file is executed prior to running the simulation. It sets up simulation 
parameters such as vectors with the rocket's aerodynamic coefficients (functions of Mach) 
and time profiles for mass properties and the main rocket thrust. The file also sets the initial 
conditions for all the simulation states. When run, the simulation uses a user-selectable 



1 Silicon Graphics, Inc. Indigo II workstation 



51 



integration scheme to simulate the rocket flight. The simulation writes several output 
vectors to the Matlab workspace. These outputs include the flight time, target miss 
distance, the guided rocket's states (U, V, W, P, Q, R, North, East, Altitude, (p, 6, and 
y/), and their time derivatives with respect to the appropriate coordinate frames. 

Figure 25 shows a top-level block diagram of the HRGK 6DOF. Figure 26 and 
Figure 27 respectively show detailed views of the "Rocket Dynamics" and the "Parameters 
and Coefficients" blocks as they are currently implemented. 



0- 



Clock 



55 



time 



flight time 



2.75" GUIDED ROCKET SIMULATION 

TgtN 



on 



0*Rlc e 



timer noise timer 



foM-' 



Guidance 

Delay 

Switch 



off 



WsT"*^. : phi 

SkVNofses ; v\ 



r... ,, 



angle 
Off Bore 




TgtE 



Tgth 



Target 
Dynamics 



' Sj 



Angle 



'•j 



Thruster 
Cmds 



Controller 



Seeker / 
Filter 



's> 







MslN » 
MslE 



Mslh, 



phi ski i 



theta , 



psi 



V 



LOS's 



True Horiz Line of Sight 



Wind Gust Models 



Rocket True Geometry 
Dynamics 



True Vert Line of Sight 



| Dmiss | 
Miss Dist 



Demux 



Figure 25. HRGK 6DOF Simulation Overall Block Diagram. (Dashed blocks are not yet fully 
implemented.) 

The blocks in Figure 25 with dashed lines are not yet fully implemented. For 
example, the thruster control inputs are not currently generated based on seeker inputs, but 
rather, the thruster commands must be "hard coded" into the conttoller block. 

Inside the Rocket Dynamics block, the time derivatives of the rocket's states are 
computed by a Matlab function. The state derivatives are then integrated and feedback into 



52 



the function and other parts of the simulation. The rocket's aerodynamic and mass property 
parameters are determined using lookup tables in the Parameters and Coefficients block. 



Thruster 1 



1 



Thruster 2 



T1 



Thruster 3 



T2 



T3 



Thrust dat 



Thrust History 



Rkt X 



Rocket State Vector 



Rocket Rate Vector 



Rkt Xdot 



Matlab 
Function 



1/s 



ROCKET DYNAMICS 



FlatEarth 
MotionEqns 



Integrator 



Mach 



■£: 



*>-T- 



Matlab 
Function 



Mach No. 



Params & Coefs 





u.v.w, 

P.Q.R 






phi 

c 






theta 


X 

E 

CD 

Q 


* 6 


-^.P_rocket 






1 






Pn 






2 






Pe 






3 





Rocket 
States 



Figure 26. Detailed View of the Rocket Dynamics Block. 











X 

Z3 
5 




















* — I Mass_dat 


I IME & MACH 


1 




Mass History 
*-| CG.dat | 
CG History 




DEPENDENT 

— DADAft/ICTCDC 


Mach 


V 


rArlAlVIt I tno 












*— | JxR_dat 

Rocket Jx 
« — | JxK_dat 


Xcp 

h 




/ 








Guidance Kit Jx 


*^ C 


IdO-Bun 


ling 




1 




4 — I Jy_dat 


^A 


Param& 
Coefs 


Jy History 
«a — | R_Pdot 


*^ 


Thrust 






X 




rocket roll accel 




A 




c 


dO-Coas 


sting 








CNa 


/ 














/ 




CMq 












Clp 































Figure 27. Detailed View of the Parameters and Coefficients Block. 



53 



Figure 28 shows the overall architecture for the simulation currently under 
development. Most of the components of this architecture are described in the following 
section. 





































*— » 


target motion 




Geomeiry 


Environment | 

• •• i 


— 


relative 
LOS 

translation 
matrix 

absolute 
positions 




clock 

atmosphere 

gravity 

wind 




Missue r- 




























parameters & 
coefficients 






non-linear 
equations 
of motion 










^ 


Mach 












i 


i 












I 


i i 




I 




- 


seeker 








forces & 
moments 




i 






control 
logic 




















IMU 












ii 








i 























Figure 28. Future 6DOF Architecture. 



B. 



SIMULATION COMPONENTS 



The simulation shown schematically in Figure 28 is made up several components. 
These components are very briefly described in the following subsections. 

1. Environment 

The environment block simulates the environment in which the rocket operates. 
This block produces the simulation time and provides constant gravity. It also computes the 
atmospheric properties needed to calculate the Mach number and dynamic pressure. The 
block uses the Dryden wind gust model to produce wind gusts in pitch and yaw. 

2. Missile 

The missile block simulates the dynamics of the guided rocket. The block has 
several sub-components. These blocks are divided to align as closely as possible with 
actual subsystems. The missile block uses external signals and data to produce realistic 
seeker and IMU signals. From these signals control inputs are determined and applied to 



54 



the forces and moments sub-component. The forces and moments component computes all 
the body frame forces on the rocket. From these forces and moments the dynamics of the 
body can be determined by the flat-earth equations of motion. 

3. Target Dynamics 

The target block provides the target location and motion to the relative geometry 
block. This data allows the line of sight (between the rocket and the target) and the line of 
sight rate data to be calculated. 

4. Relative and Absolute Geometry 

The relative geometry block uses the target and rocket dynamics to solve the 
navigation equations (described in the following section). From the navigation equations, 
the absolute and relative position of the rocket and target can be can be computed. The line 
of sight and line of sight rates are also provided by the geometry block. 

C. EQUATIONS OF MOTION 

This section contains a brief derivation of the flat-earth equations of motion as they 
apply specifically to the HRGK 6DOF. The derivation is adapted from Stevens and Lewis 
[Ref. 15]. 

1. General Matrix-Form Equations 

Starting with generic flat earth equations of motion: 

F 

v B = -£2 B v g + B B g' Q + — (Force Equation) 

m 

cog = —J~ QgJ(dg + J~ T B + H R (Moment Equation) 

6 = E(Q>)(£> B (Attitude Equation) 

T 

Pned = ^B y B (Navigation Equation) 

where the v B is the velocity vector, [U V W] T ; gg is the gravity vector at the surface of the 

earth (adjusted for the earth rotation), [0 9.81m/s] T ; F B is the force vector, [F x F y FJ T ; 
co B is the angular rate vector, [P Q R] T ; T B is the torque vector, [L M N] T ; H Ri to be 
defined later, is a vector to account for the gyroscopic effects of the spinning rocket motor; 
O is the Euler angle vector, [<J) 6 V|/] T ; p NED is the north-east-down frame position vector, 
[p N p E p D ] T ; Qq is the body frame angular rates cross-product matrix, 



55 



n B = 



B B is the rotation matrix from the North-East-Down frame to the body frame, 






-R 


Q~ 


R 





-p 


-Q 


p 


o _ 



B B = 



10 

cos0 sin</> 



cos0 -sin0 
1 



-sin0 cos0_||_sin0 cos0 J 
J is the inertia matrix (with symmetry assumed about the x-axis), 



cosi/a sini/f 
-sini/f cosy/ 
1 



J = 

















/, 



and the Euler angle function is 



£(0) = 



1 tan0sin0 tan#cos0 

cos0 -sin<p 

sin0 cos(f> 



cosO cos 6 

The state vector is [v B T co B T F T p NED T ] T . Now the forces and moments need to be defined 
for the HRGK case. 

2. Forces and Moments 

The aerodynamic forces and moments are functions of Mach number and can be 
expressed by the coefficients and their derivatives. The following summarizes the forces 
and moments computed in HRGK 6DOF: 



/ 



F x =T-qA 



r 2 



C D + 

V "a 



F y = T y +qA(-C Na P) 



(Axial Force) 

(Side Force) 
(Down Force) 



(C, d \ 



L = qA d 



\ 113 



+ T R (Rolling Moment on HRGK) 



56 



M = T Z X T + qAd 



N = T y X T + qAd 



Cm OCXrp + 



'N, 



CP 



C M a d 

— —Q 

w 



\ 



Cm d 

Cm uXcp H R 



\ 



(Pitching Moment) 



(Yawing Moment) 



where T is the rocket motor thrust; T y and T z are the y and z components of the reaction 
thruster forces, T z = r,cos0 + T 2 cos(<(h-2/3k) + r 3 cos(</H-4/37c) and T = T^siiKp + 
r 2 sin( 0+2/371) + r 3 sin(0+4/37t); and T R is the rolling torque exerted on the HRGK by the 
spinning rocket motor, 

T R = id XR P R 

where fi is the coefficient of friction for the bearings, Jx is the axial moment of inertia for 
the rocket alone, and P R is the rocket motor's spin axis angular acceleration (adapted from 
Ref. [11]). The state P R has to be added to the state vector, and the gyroscopic affect of the 
spinning rocket is added to the moment equation. The effect can be accounted for by adding 
the vector H R to the right-hand side of the moment equation, where H R is 

T 



H.= 







,RJ x B P R 



Q'x.Pr 



3. HRGK Scalar Flat-Earth Equations 

From the general matrix equations, the HRGK-specific scalar form of the equations 
can be written in terms of the defined forces and moments. 

a. Force Equations 

U = RV-QW-g sine + ^ L 

m 



V = -RU + PW + g sm<l)cosd + ^- 

m 



W = QU-PV + g cos0cos0 + -^ 

m 



Moment Equations 






57 



Q = 


( j) 


R = 


' L A 




A ) 



PR 



J.. 7. 



RP D 



N J x 

Jy Jy 



Kinematics Equations 

0= P + tan 0(0 sin + /? cos 0) 



= QcosQ- R sirup 

,COS0 



i// = Qsirut) + R 



cos 6 



d. Navigation Equations 

P N = £/cos#cosyf + V (sin sin 6 cost// -cos sin y/) + W(cos0sin0cosy/ , + sin0siny/') 

/^ = £/cos0siny/' + V(sin0sin0sini// + cos0cosy/) + W(cos0sin0siny/-sin0cosi//') 
P D = -Usin6+ Vsin0cos# + Wcos0cos0 



D. 



NPS HRGK RESULTS 



The force and moment equations, as well as the equations of motion, are 
implemented in a Matlab function which is used by the HRGK 6DOF simulation. The 
following subsections show two examples of how the simulation was used in the design. 

1. Ballistic Flight and Maximum Range (3DOF) 

Early in during the NPS HRGK design process, the HRGK simulation was 
simplified to only include the three dimensions of the rocket's longitudinal dynamics. (The 
gyroscopic effects of the spinning rocket were also ignored.) Using the simplified 3DOF, 
the results shown in Figure 29 were obtained. These results confirmed that the HRGK- 
equipped rocket could meet the time to target and the maximum range requirements. 

2. Pulse Width Modulated Turn Rate (6DOF) 

Later in the design, the affect of thruster pulse width modulation was investigated 
using the HRGK 6DOF. Two simulations were run. The first simulated the missile flight 



58 




Figure 29. HRGK-Equipped Rocket Ballistic Trajectories. Ground launch at sea 
level with various launch elevation angles. Solid lines are for cold rocket motor, 
dashed lines are for hot rocket motor, and flight times are indicated. 

with a constant thruster force from the appropriate side only. The second used a pulsed 
sequence of all three thrusters, but with a longer pulse on the appropriate side. The results 
for the two runs are shown in Figure 30. From the analysis, it was determined that constant 
thrust does not generate satisfactory maneuvers. (In fact, in the simulation, the rocket 
began "cartwheels" without making substantial changes in cross-track). On the other hand, 
the pulsed control generated an adequate maneuver while maintaining a somewhat 
reasonable side slip angle, /?. 



















X-Trk=12.1 m 






Constant Thruster Maneuver 


$ "$ 


// 11 




to 

c 


n r\ r\ j. 


it -^i— ^— ^ 


a 


' a 







"t, C?'"' W "» 


CO 

CL 




p = 9.7° p = 29.2° 


p = 52.1° 


P = 79.7° 




o 

CO 

h- 












co 












o 


Pulsed Thruster Maneuver 




X-Trk = 13.6 m 




O 


(with short opposing pulses) 


P = 


= 32.3° 




p = 3.6° P = 12.1° P = 22.3° 


50 100 


1 50 


200 






Down -Track Range (m) 







Figure 30. Sustained and Pulsed Thruster Maneuvers. Flight paths and headings indicated for 
a 0.8 second maneuver. 



59 



E. OBSERVATIONS 

As designs become more refined and leave the conceptual phases, high fidelity 
simulations becomes more important. In the case of the NPS HRGK, the 6DOF simulation 
was helpful in performing some detailed analyses. Nevertheless, the time required per 
simulation run precluded the use of the simulation in many cases. 

Fast, modular simulations that can be easily modified for various designs are 
desirable design tools. In many cases, lower-order simulations (3DOF or 5DOF) meet all 
the needs of the designer. For example, the 3DOF version of the HRGK simulation had 
sufficient fidelity to be used in the maximum range and time to target analyses. 
Unfortunately, other analyses required a 6DOF so that the roll of the seeker and the 
gyroscopic effects of the spinning rocket could be observed. 

If the restructuring of the HRGK simulation provides the expected improvement in 
run time, this simulation would become a much more useful design tool. With run times of 
only a minute or so, several design iterations could be evaluated quickly. 



60 



V. CONCLUSIONS AND RECOMMENDATIONS 

The purpose of this thesis is to provide documentation for the several computer 
codes developed by the author during the design and analysis of the NPS HRGK. The 
thesis is also intended to provide insights concerning the use of modeling in the design and 
analysis processes. Several germane points can be made in conclusion. 

First, models with differing levels of complexity are appropriate during different 
phases of the design. Models used in the analysis of conceptual and preliminary designs are 
usually the simplest while those models used in design refinement are more complicated. 
COEA-related modeling is essential throughout the design process to ensure that the design 
meets the system's cost effectiveness goals. 

Second, simplistic models can be used to ensure the flow down of system 
requirements into design specifications. In this way, the models help ensure a design 
driven by the requirements. Often these simple models provide great insight into additional 
aspects of the design problem allowing several critical specifications to be defined in the 
earliest phases of the design. 

Third, modeling in the design process should include the appropriate mix of 
probabilistic and worst-case conditions. These two types of models can provide very 
different results which must be judiciously evaluated. 

Finally, knowing the sensitivity of a model to variations in parameters is critical to 
the proper use of the model in decision making. Sensitivity analyses or "what-if ' studies 
must be performed to ensure the modeling is providing reliable data. 

In addition to these concluding points several recommendations for follow-on work 
are also appropriate. First, the design and analysis of the NPS HRGK would have been 
facilitated by simple Matlab codes that allow the early aerodynamic sizing of the kit. Such 
a program would be helpful in evaluating the effect of nose shape and canard size and 
shape on the range of the HRGK-equipped rocket. 

Additional work is also needed to include a probabilistic target motion model in the 
maneuverability analysis code. The development of models that evaluate the required pitch 
maneuverability and the required elevation field of view would also be useful. 

Finally, the modification of the HRGK simulation to make it a generic 6DOF 
simulation would be very useful for future missile design work. The Simulink architecture 
can offer fast modular simulations that can be easily modified to meet future users' needs. 



61 



62 



APPENDIX A. MISSILE AERODYNAMICS NOMENCLATURE 

Many of the aerodynamic coefficients, stability derivatives, and various other terms and 
symbols used in this thesis are defined below. The nomenclature closely follows that used 
by Chin [Ref. 16] with some adaptations like that used by Blakelock [Ref. 17] and by 
Howard [Ref. 18]. It is worth noting two significant differences between terms associated 
with missiles and those associated with traditional aircraft. First, most missile coefficients 
are referenced to the missile's diameter or its cross-sectional area (normal to the 
longitudinal axis) rather than a wing chord or a planform area. Second, normal and axial 
force coefficients, C^ and C^, (aligned to the body-axis system) are used rather than the 
lift and drag force coefficients, Cl and Cr> (in the stability axis system). When the 
missile's angle of attack (a) is small there is very little difference between the two systems, 
but for significant angles of attack, the relationship given in the following equations should 
be used. 

Ca = Cd cos a - Cl sin a 
C N = C L cos a - C D sin a 
Reference Dimensions 

A or S n Cross-sectional area of missile body (5.94 in 2 ) 

d Diameter of missile body (2.75 in) 

L Missile length (68.75 in) 

X cg Distance from nose tip to center of gravity on missile (27 to 3 1 .7 in) 

Fli ght Conditions 

a angle of attack 

P side slip angle 

M Freestream Mach number 

V t Freestream air speed 

p Freestream air density 

q Dynamic pressure ( 4 pV t ) 



63 



Aerodynamic Coefficients* 

C T Coefficient of lift, 

qs n 

N 
C N Normal force (N) coefficient, 



q s n 

M 

Cm Pitching moment (M) coefficient, 



qS n d 
C a Axial force (A) coefficient, 

q s n 

C Dn Drag force coefficient at zero lift 

Cjxj Coefficient of drag due to normal force (D N ), — — 

Static and Dynamic Derivatives * 

op 

Cw Normal force coefficient per radian of angle of attack, — — 
a da 

op 

C M Pitching moment coefficient per radian of angle of attack, — — 
a da 

— — -M 
d J da 

Missile "States" 

£/, V, W Body axis velocities (jc, v, and z-axes) 

P, (>,/? Body-axis angular rates (about the x, y, and z-axes) 

0, 9, i// Euler Angles (roll, pitch, yaw) 

p n> p b p d Flat Earth Positions (North, East, Down) 

h height {-P D ) 



Directional coefficients and derivatives {e.g.; C Y , C^, C Y , C ^ , C ^ ) are assumed equal to their 
longitudinal counterparts {e.g.; C N , C M , C Not , C Ma , C Mq ; respectively). 



64 



APPENDIX B. NOTES ON MATLAB AND SIMULINK 



A. MATLAB 

Matlab is a high-performance numeric computation and visualization computer 
software package. It is essentially a high-level programming language that is especially 
tailored for the manipulation of matrices, (see Matlab Reference Manual and Matlab 
User 's Guide) 

Matlab commands can either be invoked in an interactive mode at the keyboard or 
through Matlab script files. Another type of file is the Matlab function. Both the script 
and function files are labeled with an "m" extension at the end of the file name and are 
commonly called M-files. 

A script M-file can run as a stand alone "batch" file invoking Matlab commands 
when the file is executed. On the other hand, a function M-file is called from within script 
files or from the keyboard. A function can be called with or without arguments and can 
return computed values when completed. 

Variables declared or set in a script file or at the keyboard are available throughout 
the Matlab "workspace." These variables are common and can be accessed by other script 
files. On the other hand, variables declared in a function M-file are local variables and are 
not available in the workspace. Furthermore, functions do not have access to workspace 
variables unless the variables are passed to the function or are declared as "global" in both 
the workspace and in the function. 

The standard syntax used in the first line of a function M-file is 

function y = funciu) 
where "function" is the Matlab function indicator, y is the variable that will be passed 
back, u is the input, and func is the name of the function (func.m). 

Files that are written in C or FORTRAN can be converted into executable Matlab 
files with a ".mex" extension by using the "cmex" or "fmex" commands, (see Matlab 
External Interface Guide) Mex-files have the advantage of running faster than normal M- 
files. 



65 



B. SIMULINK 

Simulink is a dynamic system simulation software package and an extension to 
Matlab. Simulink offers many ready-made blocks that can be assembled graphically to 
model a dynamic system. Two special Simulink blocks allow tremendous flexibility in 
modeling systems, (see Simulink User's Guide) 

The first is the Matlab function block. This block when placed in a block diagram 
will execute the commands in the indicated M-file function. So for example, the equations 
of motion for a system could be placed in a M-file function. This function would have 
inputs from simulink that include the states and the control input. The function would then 
output the states' time derivatives which could then be integrated and then passed back the 
Matlab function. 

This method runs extremely slow! 

The second more useful block is the S-function block. This block calls either a M- 
file or a Mex-file that is specifically structured to run efficiently in Simulink (see Simulink 
1.3 Release Notes). Chapter 4 of Simulink 1.3 Release Notes outline the procedure for 
creating Mex-files from C codes for use in fast running Simulink models. 

Additionally, simulink models that do not contain M-file references (that use Mex- 
files in the S-function blocks) can be turned into stand-alone, accelerated, executable code 
using Simulink's Real-Time Workshop (see Real-Time Workshop User's Guide). 



66 



APPENDIX C. MATLAB CODE LISTINGS 



This appendix provides listings of the following Matlab computer codes which 
were discussed in this thesis: 

• Single-Shot Probability of Kill 

• Length, Weight, and Center of Gravity 

• Turn Rate and Radius 

• Steady-Turn Maneuver 

• Seeker Field of View 

• 6DOF Codes 
-Initialization Code 
-Aerodynamics Data 
-Equations of Motion 
-Line of Sight 
-Mach 

The user-defined inputs in the codes are denoted with bold-face type. 
C. SINGLE-SHOT PROBABILITY OF KILL 

% Single Shot Pk using "Cookie Cutter" methodology 

% W. Mark Wonnacott, Naval Postgraduate School, May 1997, corrected/revised Aug 1997 

n%%%%nnHn%%iNPOTs 

% Terminal Dive Angle (deg relative to horizontal) 

dive « 20; 

% CEPs to be used in computations (for a single CEP set CEP_min egual to CEP_max) 

CEP_min = .5; % minimum CEP (m) to be used 

CEP.max = 10 0; % maximum CEP (m) to be used 

% Target Size (m) , Prob. of Kill given Hit, & Desired Prob. Kill 



trgt='Light Arm. (AirDef)' 
trgt=' Light Armor (APC) ' ; 
trgt= ' Large Truck ' ; 



L = 5.5; H = 2.0; W = 2.5; Pkh =.90; PkD =.95 

L = 5.5; H = 2.0; W = 2.5; Pkh =.80; PkD =.60 

L = 8.0; H = 2.5; W = 2.5; Pkh = 1; PkD =.50 

L = 14.; H = 1.5; W = 4.5; Pkh =.70; PkD =.75; trgt='Patrol Boat' 

%%%%%%%%%%%%%%%!. NP UTS 

% Elevation and Crossrange Projected Dimensions (l:Head on & 2:Broadside) 

Yl = H*cos(dive*pi/180) +L*sin(dive*pi/180) ; 

XI = W; 

Y2 = H*cos(dive*pi/180)+W*sin(dive*pi/180) ; 

X2 = L; 

%CEPs in desired range 

if CEP_min<=0, CEP_min=eps; end %CEP_min set to allowable small number 
if CEP_max<=0, CEP_max=eps ; end %CEP_max set to allowable small number 
if CEP_min>CEP_max, temp=CEP_min; CEP_max=CEP_min; CEP_min=temp; end 



67 



CEPs = logspacedoglO (CEP_min) , loglO (CEP_max) ) ; 

sigmas = CEPs/1 .1774; %corresponding sigmas, 1-D miss distances (m) 

S = sigmas*sqrt (2) *2; %normalizd half sigmas for Matlab 

% Probability of Hit (equal (50%) prob. of broadside or head-on encounter) 

Ph = 0.5*(erf (Yl./S) .*erf (Xl./S) ) + 0.5* (erf (Y2. /S) . *erf (X2 . /S) ) ; 

% Probability of Single Shot Kill 

PkSS = Pkh*Ph; 

% Number of Weapons Needed to Achieve Desired Prob. of Kill 
N = log(l-PkD) ./log(l-PkSS) ; 

figure (1) % Single-Shot Probability of Kill 

semilogx (CEPs, PkSS, '-') , grid on 

axis ( [CEP_min, CEP_max ,0,1]) 

xlabeK'CEP (m) ' ) , y label (' Single Shot Probability of Kill') 

figure (2) % Number of Weapons per Kill 

loglog (CEPs,N, ' -' ) , grid on 

axis( [CEP_min,CEP_max, 1,1000] ) 

xlabeK'CEP (m) ' ) , ylabel ([' Number of Rockets for Pk = ' ,num2str (PkD) ] ) 

figure (3) % Relative Cost for Equal Cost/Kill 

Nc = max (1, ceil (N) ) ; 
[X,Y]=stairs(CEPs,Nc(D ./Nc) ; 
plot (X,Y, '-' ) , grid on 
axis ( [CEP_min, CEP_max, 0,1]) 
xlabel ( 'CEP (m) ' ) , ylabel ( 'Relative Weapon Cost' ) 

D. LENGTH, WEIGHT, AND CENTER OF GRAVITY 

% GUIDED Rocket Center of Gravity as a Function of Kit Weight 
%============================================================ 

% W. Mark Wonnacott, Naval Postgraduate School, May 1997 

%%%% Design Space Paramters 

Wkl = linspace ( 1, 8 ) ; % assummed kit mass range (lbm) 

Lkl = [5,10, 13.625,Lmax-Lr] ; % specific kit lengths (in) 

Dkl = [ . 05, . 07, . 0865, . 1, . 15] ; % specific kit densities (lbm/in~3) 

Cgkl= 1/3; % kit Cg from nose (fraction of kit length) 

dia = 2.75; % kit diameter (in) 

area> pi* (dia/2 ) A 2; % kit cross section area (in~2) 



68 



%%%% Constraint Parameters 

Lmax= 72; % max total length 

Wmax= 30; % max total weight 

CgL = 1/3; %pod forward Cg limit (frac of lug spacing back from front lug) 



%%%% Rocket Parameters 
Lr = 55.125; 
CgR_live = 29.96; 
CgR_burn = 33.55; 
WR_livo = 22.95; 
WR_burn = 15.73; 



% rocket length 

% live rocket Cg (inches from base) 
% fired rocket Cg (inches from base) 
% live rocket mass (lbm) 
% fired rocket mass (lbm) 



%%%% Launcher Pod Parameters 

Llug7 = 43.2; 

Llugl9= 43.4; 

Cg7 = Llug7-(CgL)*14; 

Cgl9=Llugl9- (CgL) *14; 

Cgpod = 32.6; 

Wpod7 = 196.2, 

Wpodl9= 516.0; 



% front lug from back of launcher— 7 (in) 

% front lug from back of launcher— 19 (in) 

% constraint cg (e.g. within middle half of lug space) 

% constraint cg (e.g. within middle half of lug space) 

% loaded pod Cg from back w/o kits (in) 

% loaded pod (7) mass w/o kits (lbm) 

% loaded pod (19) mass w/o kits (lbm) 



% (VARIOUS DENSITY CURVES PLOTTED) 
% ================================ 

Wk = Wkl' *ones(l, length (Dkl) ) ; % matrix of kit masses 

Lk = Wk(: ,l)*(l./(area*Dkl) ) ; % matrix of kit lengths (in) 

Cgk= Lr+(1-Cgkl) *Lk; % kit Cg locations (inches from base) 



Mk = Wk.*Cgk; 



% Moment about base due to kit (lbm in) 



% centers of gravity are calculated from base in inches 
Cg_live = (WR_live*CgR_live+Mk) ./ (WR_live+Wk) ; 
Cg_burn = (WR_burn*CgR_burn+Mk) . / (WR_burn+Wk) ; 

% centers of gravity are changed to calibers from nose 
Cg_live = ( (Lr+Lk) -Cg_live) /dia; 
Cg_burn = ( (Lr+Lk) -Cg_burn) /dia; 

%figure 
figure (1) , 

plot (Wk(:,l), Cg_live) , grid on 

title ('Live Round with Kit') 

ylabel ('Cg (Calibers from nose) ' ) 

xlabel ('kit weight (lbm)') 

hold on 



69 



figure (2) , 

plot (Wk( : , 1) , Cg_burn) , grid on 
title ("Fired Round with Kit') 
ylabel ('Cg (Calibers from nose) ' ) 
xlabel ('kit weight (lbm)") 
hold on 



% (VARIOUS LENGTH CURVES PLOTTED) 
% =============================== 

Lk = Lkl; 

Dmin = 0. 8*min(Dkl) ; 

Dmax = 1.2*max(Dkl) ; 

Dk = linspace (Dmin, Dmax) ; 

Dk = Dk ' *ones ( 1 , length ( Lk) ) ; 

Wk = area*Dk( : , l)*Lk; 

Cgk= Lr+(l-Cgkl)*Lk; 

Cgk= ones ( length (Wk) , 1 ) *Cgk; 



% specific kit lengths (in) 

% lower limit on assumed density range 

% upper limit on assumed density range 

% assummed kit density range (lbm/in"3) 

% matrix of kit densities 

% matrix of kit masses (lbm) 

% kit Cg locations (inches from base) 

% matrix of Cg locations 



Mk = Wk.*Cgk; 



% Moment about base due to kit (lbm in) 



% centers of gravity are calculated from base in inches 
Cg_live = (WR_live*CgR_live+Mk) . / (WR_live+Wk) ; 
Cg_burn = (WR_burn*CgR_burn+Mk) . / (WR_burn+Wk) ; 

% centers of gravity are changed to calibers from nose 
Cg_live = ( (Lr+ones (length (Wk) , 1) *Lk) -Cg_live) /dia; 
Cgjourn = ( (Lr+ones ( length (Wk) , 1) *Lk) -Cg_burn) /dia; 

figure (1) , 

plot (Wk, Cg_live) 

plot ([ (Wmax-WR_live) *ones(l,2) ], [1,20] ) %max weight constraint 
figure (2), 

plot (Wk , Cg_burn) 

plot ( [ (Wmax-WR_live) *ones(l,2) ] , [1,20] ) %max weight constraint 

% (POD CENTER OF GRAVITY CONSTRAINT) 
% ================================== 

Wk = Wkl ' *ones (1, 2) ; % matrix of kit masses 

%length from loaded pod (w/o kits) cg to constraint Cg (in) 

LI = ones ( length (Wk) , 1) * ( [Cg7,Cgl9] -Cgpod) ; 

% loaded pod weights (lbm) 

Wp = ones ( length (Wk) , 1) * [Wpod7,Wpodl9] ; 

X = ones (length (Wk) ,1) * [7, 19] ; % number of kits per pod 

% length from rocket nose back to front lug 

% (assume rocket base 4" from back of pod) 

Lt = ones ( length (Wk) , 1) * [Lr-Llug7+4,Lr-Llugl9+4] ; 



70 



% Solve for allowable kit length 

Lk = (1/ (1-Cgkl) ) * ( (LI. *Wp) ./ (X.*Wk)-Lt) ; 

Cgk= Lr+ (1-Cgkl) *Lk; % kit Cg locations (inches from base) 

% center of gravity calculated from base in inches 
Cg_live = ( (WR_live*CgR_live) + (Cgk. *Wk) ) . / (WR_live+Wk) ; 
Cg_burn = ( (WR_burn*CgR_burn) + (Cgk . *Wk) ) . / (WR_burn+Wk) ; 

% center of gravity changed to calibers from nose 
Cg_live = ( (Lr+Lk) -Cg_live) /dia; 
Cg_burn = ( (Lr+Lk) -Cg_burn) /dia; 

figure ( 1) , 

plot(Wk,Cg_live) 

axis([1.8,7.2,7.8,14]) 

%hold off 
figure (2) , 

plot (Wk,Cg_burn) 

axis( [1.8,7.2,6.8,13] ) 

%hold off 

E. STEADY-TURN MANEUVER 

% This m-file computes thrust required, T, for certain load factor, n 

% (g's) , in a steady turn for various Mach numbers. The load factor is 

% achieved using a thruster forward of the CG. 

% 

% The inputs include: 

% Scenario Definition: 

% n: desired load factor (g's) 

% rt : desired turn radius (ft) 

% alt: altitude (ft) (std day assumed) 

% Mmin: min Mach number 

% Mmax : max Mach number 

% Rocket Configuration: 

% Wght : rocket mass (lbm) 

% Dia: rocket cross-section diameter (in) (for ref area and length) 

% Xcg: rocket center of gravity in calibers from nose 

% Xt : thruster location in calibers from nose 

% Aerodynamic Parameters: 

% CNa : normal force coefficient per radian of alpha [f(Mach)] 

% Xcp : center of pressure location in calibers from nose [f(Mach)] 

% CMq: moment coefficient per rad/sec of pitch rate [f(Mach)] 

% (Aero Paramaters are linearly interpolated between points in table) 



71 



% Code uses normal force and pitch moment equations: 

% Fz=nW=N+T= (Qbar) *S*CNa*alpha + T, and 

% m = = -N*(Xcp-Xcg) + T*(Xcg-Xt) - CMq*q* (Dia/2V) *Qbar*S*Dia 

% {since, q = theta-dot = V/R = V/(V /v 2/n) = n/V 

% and N = (Qbar)*S*CNa*alpha} 

% M =(Qbar*S*CNa*alpha) (Xcp-Xcg) + T* (Xcg-Xt) - CMq*n* (Dia~2/V~2) *Qbar*S 

% The two equations are solved for the unknowns, alpha and T, using several 

% values of Mach number. 

% W. Mark Wonnacott, Naval Postgraduate School, May 1997 
clear, close all 



% INPUTS 



% 



% Scenario Definition 

n = 15; 

Rt = 6000; 

req_type = 

alt = 500; 

Mmin= 0.4; 

Mmax= 1.6; 



2; 



% desired load factor (g's) 
% desired min turn radius (ft) 
% requirement (l:turn rate, 2: turn radius) 
% std day pressure altitude (ft MSL) 
% minimum Mach number for calculations 
% maximum Mach number for calculations 
% Rocket Configuration (based on burned-out, 30 lbm, 68.75" rocket) 
Wght = 22.78; % rocket mass (lbm) 

Dia = 2.75; % reference length (in) 

Xcff =10.0; % CG in calibers from nose 

Xt = 2.2; % thruster location in calibers from nose 

% Aerodynamic Parameters (based on final config 68.75"-w/ 3 case#8 strakes) 



% 

aero table = 



Mach# 
[0 

.6614 
866 
9682 



0308 
118 
1.25 
1.4142 
1.6008 
1.8028 
2.0156 

Mach_dat= aero_table ( : , 1 ) ; 

CNa_dat = aero_table ( : , 2 ) ; 

Xcp_dat = 1 . 03 *aero_table ( 

CMq_dat = aero_table ( : , 4 ) ; 



CNa Xcp CMq 

15.0616 11.52547 -4166.814 
16.41589 11.32152 -4556.4946 
18.10928 11.01214 -5041.0303 
20.04303 10.36514 -5518.2495 
20.05156 10.35951 -5519.5352 
20.05156 10.35951 -5519.5352 
18.91632 10.45431 -4859.3901 
17.51701 10.5577 -4376.4146 
16.10659 10.68451 -3938.8662 
14.68889 10.88046 -3541.0081 
13.47025 10.99184 -3202.1553 
12.38573 11.05396 -2916.2678]; 

% Mach numbers 

% Body Normal Force per Radian Alpha 
,3); % Center of Pressure in calibers from nose 

% Pitching Moment per rad/sec of Pitch Rate 



72 



% CONSTANTS 
gO = 32.17; 
TempO = 518.67; 
PresO = 14.696*144; 
B_atm = 0.003566; 
R_air = 53.35; 
k_air = 1.4; 



% Gravity Constant. 
%surface temperature (°R) 
%surface pressure (lbf/ft' > 2) 
%temp lapse rate (°R/ft) 
%gas constant (ft lb/(lbm °R) ) 
%specific heat ratio 



% INTERMEDIATE CALCULATIONS 

% Speed of Sound 

temp = TempO -B_atm*alt; 

pres = PresO* ( 1-B_atm*alt /TempO )" (1/R_air/B_ 

rho = pres/ (R_air*temp) ; 

Vs = sqrt(k_air*R_air*temp*gO) ; 

% Rocket Config 

d = Dia/12; % 

S = pi*d~2/4; % 

XT = (Xcg -Xt)*d; % 

XP_dat = (Xcp_dat-Xcg)*d; % 

Fz = n * Wght; % 

% Mach numbers and speeds 

% adjust min and max Mach to table values 
Mmin = max (min (Mach_dat ), Mmin) ; % 

Mmax = min(max(Mach_dat) ,Mmax) ; % 

kmax =50; % 

Mach = linspace (Mmin, Mmax, kmax) ; % 



% temperature at altitude (°R) 
atm) ; %pressure @ alt (lb/ft"2) 

% density at altitude (lbm/ft~3) 
% speed of sound (ft/sec) 

reference length (ft) 

reference area (ft~2) 

thruster location from CG (ft) 

center of pressure location from CG (ft) 

normal force (lb) 



min Mach number to be used 
max Mach number to be used 
number of Mach numbers to be used 
vector of Mach numbers to be used 



% THRUST AND ALPHA CALCULATIONS FOR EACH MACH NUMBER 
for k = 1 : kmax 
V = Mach(k)*Vs; 
OS = S * l/(2*gO)*rho*V"2; 
CNa = interpl(Mach_dat,CNa_dat,Mach(k) ) 
Xcp = interpl(Mach_dat,XP_dat,Mach(k) ) ; 
CMq = interpl(Mach_dat,CMq_dat,Mach(k) ) 
A = [QS*CNa, 1; 

-QS*CNa*Xcp, XT] ; 
if req_type == 2, 

B= [(V~2*Wght)/(gO*Rt) ; 
(1/2) *QS*CMq*d~2/Rt] ; 
else 

B= [Fz; (l/2)*QS*CMq*n*gO*(d/V)"2] ; 
end 

X = inv(A)*B; 
a(k) = X(l)*180/pi; 
T(k) = X(2); 
end 



% rocket speed (ft/s) 

% (dyn pres-lb/ft"2) (ref area-ft"2) (lb) 

%CNa for Mach# 

%Xcp for Mach# 

%CMq for Mach# 



% RHS vector for turn radius 



% RHS vector for turn rate 

% solution vector 

% alpha in degrees 

% required thrust in pounds 



73 



subplot (2,1,1) 

plot (Mach,T, 'r ' ) , grid on 

xlabel ( ' Mach Number ' ) 

ylabeK 'Required Thrust (lbf) ' ) 

hold on 
subplot (2,1,2) 

plot (Mach, a, 'r ' ) , grid on 

xlabel ( 'Mach Number' ) 

ylabeK 'Required alpha (deg) ' ) 

hold on 

F. SEEKER FIELD OF VIEW 

% This m-file calculates the probability of a target being in the 

% seeker field of view, FOV (deg) , for for several seeker acquistion 

% ranges (Racq) and fields of view. Only single launch range is used. 

% 

% inputs include: Launcher Pointing Error, Pnt (deg) - 1 sigma 

% Missile unguided ballistic error, B (mils) - 1 sigma 

% Missile distance time profile, (km) (function of fly out time) 

% Guidance time delay, Gd (sec) 

% Time delay for target ID, td (sec) 

% Min and Max Target Speed (mph) 

% also required are: 

% Seeker acquistion ranges (km) - (vector) 

% Min and Max FOV (deg) 

% Launch Range (km) 

% Mark Wonnacott, April 1997 
% Naval Postgraduate School 
clear, close all 

%% Target Description 

VTmin =20; % (mph) min target speed 

VTmax =60; % (mph) max target speed 

%% Missile Configuration Inputs 
% Launcher Pointing Error 

Pnt =3; % (deg- 1 sig) initial pointing error 

% Unguided Missile Ballistic Error 

B = 12; % (mills- 1 sig) meters error per km flight 

% Guidance Delay 

Gd = 1.0; % (sec) max range delay (no guidance for 1 sec) 

% Delay time to ID target 

td = 5*1/8; % (sec) 5 laser pulses received at 8 Hz 



74 



% Seeker Field of View 

FOVmin = . 5; 

FOVmax = 45; 
% Seeker Acquisition Range 

Racq = [1.15,1.5,2,3,6] ; 
% Launch Range 

RL = 6; 
% Missile Speed Table 
Time = [0,0.22,0.89,1.02,1.67,... %flight time (s) 

2.52,4.96,7.91,11.24,14.88,18.8,22.99,27.3] ; 
Dist = [0,0.01,0.2,0.27,0.6,1,2,3,4,5,6,7,8]; % flight dist (m) 



% (deg) min half FoV 

% (deg) max half FoV 

% (km) parametric seeker acquistion ranges 

% (km) maximum launch range 



%% Inputs converted to usable variables 

VTmin2 = VTmin/2237; 

VTmax2 = VTmax/2237; 

PntR = Pnt*pi/180; 

Bkm = B/1000; 

kmax = length (Racq) ; 

mmax = 30; 

FOV = linspace (FOVmin, FOVmax.mmax) *pi/180; 

Rg = interpl (Time, Dist, Gd) ; 



% min target speed (mph->km/s) 

% max target speed (mph->km/s) 

% initial pointing error (deg->rad) 

% X-trk error (km per km flight) 

% number of seeker acq. ranges 

% number of FOV's 

% FOV's (deg->rad) 

% unguided flyout range (km) 



Pacq = zeros (kmax, mmax) ; 



% pre-set Pacq matrix size 



for k = 1 : kmax 
for m = l:mmax 

D = Racq(k)*cos(FOV(m) ) ; 
F = Racq (k)*sin (FOV (m) ) ; 
if RL-Rg > D 

Rd = RL - D; 
else 

Rd = RL - Rg; 
end 

tD = interpl (Dist, Time, Rd) ; 
tT = tD + td; 

Rt = interpl (Time, Dist, tT) ; 
s = Rt*sqrt (tan (PntR) A 2+(BkmP2) 
rl = tT * VTmin2; 
r2 = tT * VTmax2; 



% calculations for each Racq 
% calculations for each FOV 

% detection range (trgt @ FOV limit) 

% max x-trk offset for FOV 

% range flown before detection 

% range flown when guided flight begins 



% time at detection 

% time at start of turn 

% distance flown at start of turn 

%1 sig X-trk error at start of turn 

% min target motion at start of turn 

% max target motion at start of turn 



L5 = min(5*s, r2+F) ; % integration upper limit 

%%% Integration of Pr (Xr) *Pt (Xr-F,Xr+F) 

%%% from to L5 done using 'quad8' (Pacq function in 'ProbAcq') 
Pacq(k,m) = 2*quad8 ( 'ProbAcq' , 0,L5, 0.01, [] ,F,s,rl,r2) ; 



end 



end 



75 



FOV = FOV*180/pi; % field of view changed back to degrees 

figure 

plot (FOV, Pacq), grid on 

title (['Vt = [ ' ,num2str (VTmin) , ' -> ' , num2str (VTmax) , ' ] mph']) 

legend (num2str (Racq(l) ) ,num2str (Racq(2) ) ,num2str (Racq(3) ) , num2str (Racq(4) ) , num2str (Rac 

q(5))) 

ylabel ('Probability of Target in FOV), 

xlabeK'half FOV (±deg) ' ) 

ProbAcq Function 

function Pacq = ProbAcq(Xr,F,s,rl,r2) 

% Returns the product of the assumed probability of the rocket being 

% approximately at a certain cross-track position, Xr, and the 

% probability of the target being between ±F to either side of 

% the rocket position (seeker's FOV limits). 

% The probabilities are based on the 1-sigma rocket cross-track error— s 

% and the min and max possible target cross-track motion— rl and r2 . 

% 

% This function has to take Xr as a vector and output Pacq as a vector! ! ! 

% W. Mark Wonnacott, Naval Postgraduate School, 1997 

if rl == 0; rl = 0.0001; end % avoid division by zero 

Prkt = l/(s*sqrt(2*pi) ) *exp(-Xr.~2. / (2*s~2) ) ; %normal distribution of rocket 

% Find probability of target in FOV extremes (Pt(xl,x2)) for each Xr 
for k = 1: length (Xr) 

x = Xr(k); % step through each rocket x-trk position 

XI = x - F; % extreme bound of FOV 

X2 = x + F; % extreme bound of FOV 

if XI > r2 % weapon farther out than possible for target 

Ptgt(k) = 0; 

elseif XI < -r2 % all possible target locations inside FOV 

Ptgt(k) = 1; 

else 

% compute area bounded rl & r2 circles, fly-out axis, and X2 

LI = min(X2,r2); % limit of intgrt'n for r2 circle 

L2 = min(X2,rl); % limit of intgrt'n for rl circle 

Ala = Ll*sqrt(r2~2-Ll~2) + r2~2*asin(Ll/r2) ; %bounded area in r2 circle 

Alb = L2*sqrt(rl~2-L2~2) + rl~2*asin(L2/rl) ; %bounded area in rl circle 

Al = Ala-Alb; 



76 



% compute area bounded rl & r2 circles, fly-out axis, and XI 

L3 = min(abs(Xl) ,r2) ; 

L4 = min(abs(Xl) ,rl) ; 

A2a = L3*sqrt(r2 /S 2-L3~2) + r2 A 2*asin(L3/r2) ; 

A2b = L4*sqrt(rl"2-L4"2) + rl"2*asin(L4/rl) ; 

A2 = A2a-A2b; 

A = Al-sign(Xl)*A2; 
Ptgt(k) =A/ (pi*(r2"2-rl"2) ) ; 
end 
end 

Pacq = Prkt.*Ptgt*100; 
return, end 

G. 6DOF CODES 

1. Simulation Initilization 

% File Initializes Simulation 

clear 

data_out = [2001,5] ; 

%%% Missile INITIAL CONDITIONS (6 DOF Flight) 

%%% (u,v,w,p,q,r,phi, theta,psi,p_rocket,pn,pe,ph) 

% ground launch initial conditions 

Msl_Xo = [0;0;0;0;0;0;0;10*pi/180;0;0;0;0;0] ; 

% target pointing after 12 sec flight (10° elev launch) trgt § 5000m 

%Msl_Xo = [285;0;0;0.03; . 01; 0;0; -.383 ; 0; -125 . 6;4230;0;310] ;%theta =0.0246 

% straight and level cruise after 12 sec flight 

%Msl_Xo = [300;0;0;0;0;0;0;0;0;-125.6;0;0;300]; 
%%% Target INITIAL CONDITIONS (6 DOF Flight) 
%%% (Pn, Pe) 

Tgt_Xo = [5000, -00]; 

T_Vn = 0; 

T_Ve = 

Talt = 



Aerodat % weight and aero data file for 2.75" HRGK-equipped rocket 



77 



2. Aerodat 

% This File Contains Look-up Table Data 

% 2.75" Rocket with Final Design Guidance Kit 

global gO TempO PresO B_atm R_air S d Sd Xt mu rho c; 

%CONSTANTS & CONVERSIONS 

%%% Conversions 

in2m = 39.37; % inches per meter 

lb2kg = 2.2046; %pounds per kilogram 

%%% Gravity 

gO = 9.81; % Gravity Constant, g'0, in m/s^2. 

%%% Atmospheric Data: 

TempO = 288.15; %surface temperature (K) 

PresO = 101330; %surface pressure (Pa) 

B_atm = 0.00650; %temp lapse rate (°C/m) 

R_air = 287.0; %gas constant (N m/ (kg K) ) 

% ===================================================== 

% Guidance Package Parameters 

% ===================================================== 

radius = 2.75/2/in2m; %Guidance Package Radius (m) 

GP_Mass0 = 7/lb2kg; %Initial Guidance Package Mass (kg) 
GP_Massf = 7/lb2kg; %Final Guidance Package Mass (kg) 
GPJLngth = 13.625/in2m; %Guidance Package Length (m) 

Xt = 6.6/in2m; %Thruster Location from Nose (m) 

mu = 0.006; %Rocket to Kit Coefficient of Friction 

GP_CG0 = 8.5/in2m; %Initial Guidance Package CG from nose (m) 
GP_CGf = 8.5/in2m; %Final Guidance Package CG from nose (m) 

GP_Jxx0 = ... %Aprox. Initial Axial Moment of Inertia (kg m^2) 

GP_MassO*radius A 2/2; 
GP_Jxxf = ... %Aprox. Final Axial Moment of Inertia (kg m~2) 

GP_Massf *radius"2/2 ; 

GP_Jyy0 = ... %Aprox. Initial Transverse M. of I . (kg m~2) 

GP_Mass0*(GP_CG0-GP_Lngth/2)~2 + GP_MassO*GP_Lngth~2/12; 

GP_Jyyf = ... %Aprox. Final Transverse M. of I. (kg m"2) 

GP_Massf*(GP_CGf-GP_Lngth/2) A 2 + GP_Massf*GP_Lngth' N 2/12; 

FlyTime = 30; %Approximate Flyout Time Bound (sec) 



78 



% ===================================================== 

% Rocket Parameters from IHSP 89-289, Mk-66 Data Book 
% ===================================================== 

d = 2.75/in2m; %Rocket Diameter (m) 



R_MassO = 22.95/lb2kg; 
R_Massf = 15.73/lb2kg; 
R_Lngth = 55. 125/in2m; 



%Initial Rocket Mass (kg) 
%Final Rocket Mass (kg) 
%Rocket Length (m) 



R CGO 



R_Lngth-29.96/in2m + GP_Lngth; 



R CGf 



%Initial Rocket CG from nose (m) 



%Final Rocket CG from nose (m) 



R_Lngth-33.55/in2m + GP_Lngth; 



R_JxxO = 26.2/(lb2kg*in2nT2) ; 
R_Jxxf = 1 9 . 7 / ( lb2 kg* in2m~ 2 ) ; 



%Initial Axial Moment of Inertia (kg m^2) 
%Final Axial Moment of Inertia (kg m /v 2) 



R_JyyO = 6248/(lb2kg*in2nT2) ; 
R_Jyyf = 5008/(lb2kg*in2nr2) ; 



%Initial Transverse M. of I. (kg m^2) 
%Final Transverse M. of I. (kg m"2) 



BurnTime = 1.05; 



%Average Rocket Motor Burn Time (sec) 



Thrus 


tHot_dat= . 


[0.00 





0.05 


1600 


0.07 


1700 


0.10 


1690 


0.15 


1600 


0.20 


1530 


0.25 


1500 


0.30 


1490 


0.35 


1510 


ThrustCold_dat= 


[0.00 


0.05 


1150 


0.07 


1300 


0.1 


1380 


0.15 


1400 


0.2 


1400 


0.25 


1400 


0.3 


1390 


0.35 


1400 



0.40 
0.45 
0.50 
0.55 
0.60 
0.65 
0.70 
0.75 
0.80 



%Thrust Time History (sec & lbf for 150°F) 
1520 0.85 1800 

1560 
1610 
1650 
1700 
1730 
1750 



0.90 


1700 


0.95 


600 


1.00 


200 


1.05 


40 


1.10 


10 


1.15 





FlyTime 0] ; 



1780 
1800 

%Thrust Time History (sec & lbf for -50°F) 
0.4 1380 0.85 1680 



0. 


.45 


1350 


0. 


.5 


1340 





,55 


1350 





.6 


1350 





.65 


1400 





.7 


1440 





.75 


1500 





.8 


1590 



0.9 


1760 


0.95 


1750 


1.00 


1600 


1.05 


700 


1.1 


100 


1.15 





FlyTime 0] ; 



ThrustHot_dat ( : , 2 ) = ThrustHot_dat ( : , 2 ) *g0 /lb2kg; 
ThrustCold_dat ( : , 2 ) = ThrustCold_dat ( : , 2 ) *g0/lb2kg; 



79 



% Reference Area/Length and Mass & Moment Time Histories 
% ===================================================== 

S = pi* (d~2) /4; % Cross-Sectional Area (Ref Area-nT2) 

Sd = S*d; % Ref Area x Ref Length (m~3) 



t = [0; BurnTime; FlyTime] ; 



%time vector for time histories 



Mass_dat = [t, [R_MassO+GP_MassO 
R_Mas s f +GP_Mas s 
R_Massf+GP_Massf ] ] ; 



% Mass Time History (sec & kg) 



cg0= (R_CG0*R_Mass0+GP_CG0*GP_Mass0) / (R_MassO+GP_MassO) 
cgl= (R_CGf *R_Massf+GP_CG0*GP_Mass0) / (R_Massf+GP_MassO) 
cg2= (R_CGf *R_Massf +GP_CGf *GP_Massf ) / (R_Massf +GP_Massf ) 
CG_dat = [t, [cgO % CG History (sec & m from nose) 

cgl 

cg2]]; 



JxO = R_JxxO+GP_JxxO 
Jxl = R_Jxxf+GP_JxxO 
Jx2 = R_Jxxf+GP_Jxxf 
Jx_dat = [t, [JxO 
Jxl 
Jx2]]; 



% Axial M. of I. History (kg m~2) 



JxR_dat = [t, [R_JxxO; R_Jxxf; R_Jxxf ] ] ; 
JxK_da t = [ t , [ GP_ JxxO ; GP_ JxxO ; GP_Jxxf ] ] ; 



JyO = R_Jyy0+R_Mass0*(R_CG0-cg0)~2+GP_Jyy0+GP_Mass0*(GP_CG0-cg0)' N 2 
Jyl = R_Jyyf+R_Massf*(R_CGf-cgl)~2+GP_Jyy0+GP_Mass0*(GP_CG0-cgl)"2 
Jy2 = R_Jyyf+R_Massf*(R_CGf-cg2)"2+GP_Jyyf+GP_Massf*(GP_CGf-cg2) A 2 
Jy_dat = [t, [JyO % M. of I. History (kg m~2) 

Jyl 

Jy2]]; 



% Rocket Roll Acceleration History 
R Pdot = . . . 



[0 





0.5 


45.0 


0.6 


100.0 


0.9 


15.0 


1.0 


-50.0 


1.2 ■ 


-120.0 


1.4 


-92.0 


1.6 


-34.0 


R_Pdot ( 


:,2) = 



% Rocket Roll Accel (rev/s/s) 



1.8 


-8.0 


2.0 


0.0 


2.4 


2.0 


3.0 


7.5 


3.6 


8.5 


4.0 


9.5 


4.4 


7.5 


5.6 


0.0 



5.8 


-4.5 


6.2 


-12.5 


6.6 


-10.0 


7.0 


-7.0 


8.0 


-4.0 


9.5 


0.0 


12. 


1.0 


21. 


0.1] 



R_Pdot(: ,2)*2*pi; 



% converted to rad/s/s 



80 



% Aerodynamics Data from ADAM ' s MADR 

Mach_dat = ... 

[0 1.0000 1.4142 

0.6614 1.0308 1.6008 

0.8660 1.1180 1.8028 

0.9682 1.2500 2.0156]; 
%Static Coefficient /Derivative Components Based on Mach Number Vector 

% 

CdOCoast_dat = ... % Skin, Wave, and Base Drag (rocket coasting) 

[0.52292 0.83218 1.09629 

0.48521 1.0067 1.13922 

0.51417 1.00424 1.16393 

0.75708 1.02549 1.17799]; 

Cd0Burn_dat = . . . % Skin and Wave Drag (rocket burning) 

[0.42291 0.73219 0.90267 

0.38521 0.81307 0.95226 

0.41417 0.81061 " 0.99252 

0.65708 0.83187 1.02295]; 

CNa_dat = . . . % Body Normal Force per Radian Alpha or Beta 

[15.0616 20.05156 16.10659 

16.41589 20.05156 14.68889 

18.10928 18.91632 13.47025 

20.04303 17.51701 12.38573]; 

CP_dat = . . . % Center of Pressure in Calibers 

[11.52547 10.35951 10.68451 

11.32152 10.35951 10.88046 

11.01214 10.45431 10.99184 

10.36514 10.5577 11.05396]; 

CP_dat = CP_dat*d; 

%Dyanamic Coefficients /Derivatives Based on Mach Number Vector 

% 

CMq_dat = . . . % Pitching damping per Radian/sec of Pitch Rate 

[-4166.814 -5519.5352 -3938.8662 

-4556.4946 -5519.5352 -3541.0081 

-5041.0303 -4859.3901 -3202.1553 

-5518.2495 -4376.4146 -2916.2678]; 

Clp_dat = . . . % Roll damping (HRGK only) /Radian/sec of Roll Rate 

[-7.513 -7.513 -6.7529 

-7.513 -7.513 -6.453 

-7.513 -7.2767 -5.5802 

-7.513 -7.0348 -4.7831]; 



81 



3. Equations of Motion 

function xo = MotionEqns (uo) 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

%%% This Function computes the Flat-Earth, Body Axes, 

%%% 6-DOF Dynamics Equations for a 

%%% Thruster -Controlled Missile 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

%%% Adapted for rolling airframe, thrust control, and 

%%% variable aero coefficient and missile parameters by 

%%% W. Mark Wonnacott, March 1997 

%%% Naval Postgraduate School 

%%% Adapted from the G. Hutchins' Code for EC 4330. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

%%% Original Code by R. G. Hutchins, 10 December 1996 

%%% Thanks to 

%%% Rob King and Mark Wonnacott for aero assistance 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

%%% The State Vector is defined as: 

%%% Body-Axes Velocities 

%%% x(l) = U, speed along missile axis, x direction 

%%% x(2) = V, speed of sideslip, y direction 

%%% x(3) = W, speed along z direction ("Down") 

%%% Body-Axes Angular Velocities 

%%% x(4) = P, roll rate 

%%% x(5) = Q, pitch rate 

%%% x(6) = R, yaw rate 

%%% Euler Angles 

%%% x(7) = phi, roll relative to vertical down 

%%% x(8) = theta, pitch 

%%% x(9) = psi, yaw 

%%% Rocket Rolling 

%%% x(10) = Pr, rocket roll rate 

%%% Location Variables 

%%% x(ll) = Pn, Position North of (0,0,0) 

%%% x(12) = Pe, Position East of (0,0,0) 

%%% x(13) = h, Height 

%%% The input vector is defined as : 

%%% u(l) = Tl, thrust from thruster #1 

%%% u(2) = T2, thrust from thruster #2 

%%% u(3) = T3, thrust from thruster #3 

%%% u(4) = T, thrust through the missile x axis 



82 



%%%Other Missile Parameter Inputs (functions of Mach or time): 

%%% P(l) = mass, missile mass (kg) [f(t)] 

%%% P(2) = Xcg, missile Cg (m from nose) [f(t)] 

%%% P(3) = Xcp, center of pressure (m from nose) [f(M)] 

%%% P(4) = JxR rocket MofI about long axis (kg m~2) 

%%% P(5) = JxK kit MofI about long axis (kg m~2) 

%%% P(6) = Jy=Jz, missile Mofl-y or z axes [f(t)] 

%%% P(7) = PdotR, rocket roll accel (rad/s/s) [f (t) ] 

%%% Aero Coefficients /Derivatives (functions of Mach or time): 

%%% C(l) = CdO,zero alpha drag coefficient 

%%% C(2) = CNa, normal force /rad alpha (lift curve slope) 

%%% C(3) = CMq, pitch damping per rad/sec pitch rate 

%%% C(4) = Clp, roll damp per rad/sec roll rate (HRGK) 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

%%% Related Quantities 

Angle of Attack 
Sideslip Angle 

Resultant Vert. Thrust (up, -z dir.) 
Reultant Side Thrust (y direction) 
Missile Speed 

Air Temperature (K) [f (altitude) ] 
Air Pressure (Pa) [f (altitude) ] 
Air Density (Kg/m~3) [f (altitude) ] 
Dynamic Pressure = 1/2 rho(h) Vt*2 
missile length (m) 
missile diameter (m) (ref length) 
missile cross-sectional area(m^2) (ref area) 
rocket to kit coefficient of friction 
%%% Aerodynamic Forces 

%%% CN = CNa*alpha => N=CN*qbar*S+Tz Normal Force (=Lift) 
%%% CY = -CNa*beta => Y=CY*qbar*S+Ty Sideslip Force 
%%% CD = CdO + or 2 /CNa => D=CD*qbar*S Drag Force 
%%% Aerodynamic Moments 
%%% CLbar = Clp*P*d/ (2*U) , 

%%% => Lbar = CLbar*qbar*S*d Kit Roll Moment 

%%% CM = CNa*alpha(Xcg-Xcp) + CMq*Q*d/ (2*U) 
%%% => M = CM*qbar*S*d + Tz(Xcg-Xt) Pitch Moment 
%%% CNbar = CNa*beta(Xcg-Xcp) + CMq*Q*d/ (2*U) 
%%% => Nbar = CNbar *qbar*S*d + Ty(Xcg-Xt) Yaw Moment 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
%%% Declare Global Variables 
global gO TempO PresO B_atm R_air S d Sd Xt mu; 



%%% 


alpha, 


%%% 


beta, 


%%% 


Tz, 


%%% 


Ty, 


%%% 


vt, 


%%% 


Temp, 


%%% 


Pres, 


%%% 


Rho, 


%%% 


qbar, 


%%% 


L, 


%%% 


d, 


%%% 


S=pi*d*2/4, 


%%% 


mu, 



83 



%%% Define State Variables from Inputs 

x = uo(l:14) ; 

%%% Body-Axes Velocities 



U = x(l) 
V = x(2) 
W = x(3) 



% speed along missile axis, x 
% speed of sideslip, y direction 
% speed along z direction ("Down") 



%%% Body-Axes Angular Velocities 



P = x(4) 

Q = x(5) 

R = x(6) 
%%% Euler Angles 

phi = x(7) ; 

theta = x ( 8 ) ; 

psi = x(9) ; 
%%% Rocket Rolling 

Pr = x(10) ; 



% roll rate 
% pitch rate 
% yaw rate 

% roll 
% pitch 
% yaw 



% rocket roll rate 
%%% Location Variables 

Pn = x(ll); % Position North of (0,0,0) 
Pe = x(12); % Position East of (0,0,0) 
h = x ( 13 ) ; % Height 



%%% Define Control Variables from Inputs 

u = uo(14:17); 

Tl = u(l); % thrust from thruster #1 
T2 = u(2); % thrust from thruster #2 
T3 = u(3); % thrust from thruster #3 
T = u(4) ; % thrust through the missile long, axis 



%%% Other Missile Parameter Inputs (time or Mach dependent) : 

Param = uo(18:24) ; 

%missile mass (kg) 

%missile Cg (m from nose) 

%missile CP (m from nose) 

%rocket M of I about long axis (kg m^2) 

%kit M of I about long axis (kg m /N 2) 

%missile MofI about y or z axes (kg m~2] 

%rocket roll acceleration (rad/s/s) 

%%% Define Mach Number Dependent Coefficients/Derivatives 

C = uo(25:28) 

%zero alpha drag coefficient 
%normal force /rad alpha (lift curve slope) 
%pitch damping per rad/sec pitch rate 
%roll damping (kit only) per rad/sec roll 



mass = Param(l) ; 
Xcg = Param(2) 
Xcp = Param (3) 
JxR = Param(4) 
JxK = Param(5) 
Jy = Param ( 6 ) ; 
PdR = Param(7) 



CdO = C(l) 
CNa = C(2) 
CMq = C(3) 
Clp = C(4) 



84 



%%%%%%% Define the Forces and Moments 

%%% Related Quantities 
% speed quantities 

Vxz2 = U A 2 + VT2; 

Vt2 = Vxz2 + V A 2; 

Vt = sqrt ( Vt2 ) ; 
% atmospherics 

temp = TempO-B_atm*h; 

pres = PresO*(l-B_atm*h/TempO) A (gO/R_air/B_atm) ; 

rho = pres/ (R_air* temp) ; 

qbar = . 5*rho*Vt2; 
% control thrusts 

Ty = Tl*cos(phi) + ... 

T2*cos(phi+2/3*pi) + T3*cos (phi+4/3*pi) ; 

Tz = Tl*sin(phi) + ... 

T2*sin(phi+2/3*pi) + T3*sin(phi+4/3*pi) ; 
% angles of attack 

alpha = atan2(W,U); 

beta = atan2 (V, sqrt (Vxz2 ) ) ; 

%%% Forces 

CN = CNa*alpha; % normal force coefficient 

Fz = -qbar*S*CN - Tz; % normal force 

CY = -CNa*beta; % side force coefficient 

Fy = qbar*S*CY + Ty; % side force 

CD = CdO + (CN A 2+CY A 2) /CNa;% drag force coefficient 

Fx = T - qbar*S*CD; % drag force 

%%% Moments 

CI = Clp*P*d/(2*U+.001) ; % roll moment coefficient 
Lbar = qbar*Sd*Cl + mu*JxR*PdR; % roll moment on kit 
CM = CNa*alpha* (Xcg-Xcp) + ... %pitch moment coefficient 

CMq*Q*d/(2*U+.00D ; 
M = qbar*Sd*CM + Tz*(Xcg-Xt); % pitch moment 
Cn = -CNa*beta* (Xcg-Xcp) + . . . % yaw moment coefficient 

CMq*R*d/(2*U+.001) ; 
Nbar = qbar*Sd*Cn + Ty*(Xcg-Xt); % yaw moment 



85 



%%% Compute the Time Derivatives from Flat-Earth Equations 

%%% Force Equations 

Ud = R*V - Q*W - gO*sin (theta) + Fx/mass; 

Vd =-R*U + P*W + gO*sin(phi) *cos(theta) + Fy/mass; 

Wd = Q*U - P*V + gO*cos(phi)*cos(theta) + Fz/mass; 

%%% Moment Equations 

Pd = Lbar/JxK; 

Prd = PdR; 

Qd = (l-JxK/Jy)*P*R + M/Jy - ( JxR/Jy) *R*Pr; 

Rd = (JxK/Jy-l)*P*Q + Nbar/Jy + (JxR/Jy) *Q*Pr; 

%%% Kinematic Equations 

phid = P + tan(theta) * (Q*sin(phi) + R*cos (phi) ) ; 

thetad = Q*cos(phi) - R*sin(phi); 

psid = (Q*sin(phi) +R*cos (phi) ) /cos (theta) ; 

%%% Navigation Equations 

Pnd = U*cos (theta) *cos (psi) + ... 

V* (sin(phi) *sin(theta) *cos (psi) - cos (phi) *sin(psi) ) +.. 

W* (cos (phi) *sin(theta) *cos(psi) + sin(phi) *sin(psi) ) ; 
Ped = U*cos (theta) *sin (psi) + ... 

V* (sin(phi) *sin(theta) *sin(psi) + cos(phi) *cos (psi) ) +.. 

W* (cos (phi) *sin( theta) *sin (psi) - sin(phi) *cos (psi) ) ; 
hd = U*sin(theta) - V*sin (phi )* cos (theta) - ... 

W*cos (phi) *cos (theta) ; 

%%%%%%%%%% Define the output vector 
%%% Body-Axes Accelerations 



xo(l) = Ud 
xo(2) = Vd 
xo(3) = Wd 



% ace along missile axis,x direction 
% ace of sideslip, y direction 
% ace along z direction ("Down") 



%%% Body-Axes Angular Accelerations 



% roll ace 
% pitch ace 
% yaw ace 



xo(4) = Pd; 

xo(5) = Qd; 

xo(6) = Rd; 
%%% Euler Angle Rates 

xo(7) = phid; % d phi/dt 

xo(8) = thetad; % d theta /dt 

xo(9) = psid; % d psi/dt 

%%% Rocket Rolling Equations 

xo(10) = Prd; % rocket spin acceleration 

%%% Location Variable Rates 

xo(ll) = Pnd; % Position North of (0,0,0) rate 

xo(12) = Ped; % Position East of (0,0,0) rate 

xo(13) = hd; % Height rate 



86 



4. Line of Sight 

function yo = Lineof Sight (uo) 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

%%% This Function computes the seeker frame LOS, 

%%% from the tangent plane coordinates 

%%% W. Mark Wonnacott, May 1997 

%%% Naval Postgraduate School 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

%%% INPUTS 

%%% Position Differences Xt-Xm 

%%% u(l) = N, delta north 

%%% u(2) = E, delta east 

%%% u(3) = H, delta height 

%%% Euler Angles 

%%% u(4) = phi, seeker roll relative to vertical down 

%%% u(5) = theta, rocket pitch relative to horizontal 

%%% u(6) = psi, rocket yaw relative to north 

%%% OUTPUTS Seeker LOS 

%%% yd) = alpha, off-boresight angle 

%%% y(2) = beta, angle from seeker's ref. roll position (down) 

%%% y(3) = R, slant range to target 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

%%% Define Tangent Plane LOS Offsets 

N = uo(l); % north distance to target 

E = uo(2); % east distance to target 

D = uo(3); % down distance to target 

%%% Define Rocket Euler Angles 



phi = uo(4) 
theta = uo(5) 
psi = uo(6) 



% seeker roll relative to vertical down 
% rocket pitch relative to horizontal 
% rocket yaw relative to north 
%%% Transformation Matrices (NED to XYZ-body) 
CI = [cos (psi) 
-sin (psi) 

C2 = [cos (theta) 

sin (theta) 
C3 = [ 1 


X=C3*C2*C1* [N;E;D] ; 

Offbore = atan2 (sqrt (X(2) A 2+X(3) "2) ,X(1) ) ; 
pointing= atan2 (X(3) ,X(2) ) ; 
R = sgrt(X(l)' v 2+X(2) A 2+X(3)"2) ; 

yo(l) = Offbore; yo(2) = pointing; yo(3) = R; 



87 



sin (psi) 


0; 


cos (psi) 


0; 





1]; 





-sin (theta) ; 


1 


0; 





cos (theta) ] ; 





0; 


cos (phi) 


sin (phi) ; 


-sin (phi) 


cos (phi) ] ; 



5. Mach Number 

function M = Mach(uo) 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

%%% This function computes the Mach Number as a function of 

%%% component velocities and altitude 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

%%% W. Mark Wonnacott, March 1997 

%%% Naval Postgraduate School 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

%%% INPUTS: 

%%% Body-Axes Velocities 

%%% x(l) = U, speed along missile axis, x direction 

%%% x(2) = V, speed of sideslip, y direction 

%%% x(3) = W, speed along z direction ("Down") 

%%% x(13)= h, altitude (m) 

%%% Related Quantities 

%%% Vt, Missile Speed (m/s) 

%%% Temp, Air Temperature (K) as function of altitude 

%%% Pres, Air Pressure (Pa) as function of altitude 

%%% Rho, Air Density (Kg/m A 3) as function of altitude 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 

%%% Declare Global Variables 

global gO TempO PresO B_atm R_air S d Sd Xt; 

%%% Define Variables from Inputs 

%%% Body-Axes Velocities 

U = uo(l); % speed along missile axis, x direction 
V = uo(2); % speed of sideslip, y direction 
W = uo(3); % speed along z direction ("Down") 
h = uo ( 13 ) ; % height 

Vt = sqrt(lT2 + V"2 + VT2); 

% atmospherics 

k = 1.4; 

temp = TempO -B_atm*h; 

pres = PresO* ( 1-B_atm*h/Temp0 ) A (gO /R_air/B_atm) ; 

rho = pres/ (R_air*temp) ; 

c = sqrt (k*R_air*temp) ; 

M = Vt/c; 



88 



LIST OF REFERENCES 



1. AIAA Missile Systems Technical Committee. "Graduate Team Missile Design 
Competition, RFP: Hit-To-Kill Guided Rocket (HGR)." American Institute of 
Aeronautics and Astronautics; Reston, VA: June 1996. 

2. Wonnacott, W.M; Pomerantz, B; Silva, S.L; and Nurse, N.A. "The Three and a Half 
Rocketeers Hit-to-Kill Rocket Guidance Kit" (design proposal for AIAA graduate 
student competition). Naval Postgraduate School; Monterey, CA: June 1997. 

3. Foss, Christopher F. and Gander, Terry J., eds. Jane's Military Vehicles and 
Logistics, 1996-97. Jane's: 1996. 

4. Foss, Christopher F., ed. Jane's Armour and Artillery, 1996-97 (7th ed). Jane's: 
1996. 

5. Cullen, Tony and Foss, Christopher R., eds. Jane's Land-Based Air Defense, 1996- 
97 (9th ed). Jane's: 1996. 

6. Sharpe, Richard, Captain RN, ed. Jane's Fighting Ships, 1995-96. Jane's: 1995. 

7. Washburn, Alan. "Notes on Firing Theory." Naval Postgraduate School; Monterey, 
CA: 1983. 

8. Nicholas, Ted and Rossi, Rita, eds. US Weapon Systems Costs, 1994 (14th ed). 
Data Search Associates; Fountain Valley, CA: 1994. 

9. Tobin, Vince, Maj. U.S. Army and Mason, Pat, Capt. U.S. Army — AH-64 pilots, 
personal communications. Naval Postgraduate School; Monterey, CA: January 
1997. 

10. HYDRA-70/2.75 Inch Rocket Management Office. "Hydra-70 2.75 Inch Rocket 
System Information Handbook." US Army Armament, Munitions and Chemical 
Command; Rock Island, IL: April 1994. 

11. Ferguson, John H., Jr. and Garvey, Paul B. "2.75-Inch Rocket Motor Mark 66 Data 
Book: Characteristics and Performance Data," IHSP 89-289. Naval Ordnance 
Station; Indian Head, MD: June 1989. 

12. Hindes, John W., Jr. Using ADAM (3rd ed). Madison, AL: October, 1993. 

13. Wonnacott, W.M. "Initial Version of the Medium Range, Antiship Weapon Analysis 
Program— SeaHit," NWC TM 6823. Naval Air Warfare Center Weapons Division; 
China Lake, CA: September 1990. 

14. Advanced Systems Concepts Office. "Missile Sizing Program User's Manual." U.S. 
Army Missile Command; Huntsville, AL: March 1989. 



89 



15. Stevens, Brian L. and Lewis, Frank L. Aircraft Control and Simulation. Wiley; New 
York: 1992. 

16. Chin, S.S.; Missile Configuration Design; 1961: McGraw-Hill Book Company; New 
York. 

17. Blakelock, John H.; Automatic Control of Aircraft and Missiles; 2nd Ed; 1991: John 
Wiley & Sons, New York. 

18. Howard, Richard; AA3701: Missile Aerodynamics Class Notes; 1995: Naval 
Postgraduate School; Monterey, CA. 



90 



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